L(1,chi)_chi_is_primitive_odd_order

498 days ago by lfunction

for m in [5,13,29,53,61]: _sum_ = 0 for j in [1..(m-1)/2]: _sum_ = _sum_ + kronecker(j,m)*log(sin(j*pi/m)) print "The value of L-function of Legendre symbol modulo",m,"is", numerical_approx((2/sqrt(m))*(abs(_sum_))) 
        
The value of L-function of Legendre symbol modulo 5 is 0.430408940964004
The value of L-function of Legendre symbol modulo 13 is
0.662735391071846
The value of L-function of Legendre symbol modulo 29 is
0.611766289562307
The value of L-function of Legendre symbol modulo 53 is
0.540024945102558
The value of L-function of Legendre symbol modulo 61 is
0.938310198248835
The value of L-function of Legendre symbol modulo 5 is 0.430408940964004
The value of L-function of Legendre symbol modulo 13 is 0.662735391071846
The value of L-function of Legendre symbol modulo 29 is 0.611766289562307
The value of L-function of Legendre symbol modulo 53 is 0.540024945102558
The value of L-function of Legendre symbol modulo 61 is 0.938310198248835
for m in [5,13,17,29,37,41,53,61,73,89,97,101]: _sum_ = 0 for j in [1..(m-1)/2]: _sum_ = _sum_ + kronecker(j,m)*log(sin(j*pi/m)) print "The value of L-function of Legendre symbol modulo",m,"is", numerical_approx((2/sqrt(m))*(abs(_sum_))) 
        
The value of L-function of Legendre symbol modulo 5 is 0.430408940964004
The value of L-function of Legendre symbol modulo 13 is
0.662735391071846
The value of L-function of Legendre symbol modulo 17 is 1.01608483384284
The value of L-function of Legendre symbol modulo 29 is
0.611766289562307
The value of L-function of Legendre symbol modulo 37 is
0.819292168725432
The value of L-function of Legendre symbol modulo 41 is 1.29909306157510
The value of L-function of Legendre symbol modulo 53 is
0.540024945102558
The value of L-function of Legendre symbol modulo 61 is
0.938310198248835
The value of L-function of Legendre symbol modulo 73 is 1.79463648373515
The value of L-function of Legendre symbol modulo 89 is 1.46444140226402
The value of L-function of Legendre symbol modulo 97 is 1.89349532374748
The value of L-function of Legendre symbol modulo 101 is
0.596668668017519
The value of L-function of Legendre symbol modulo 5 is 0.430408940964004
The value of L-function of Legendre symbol modulo 13 is 0.662735391071846
The value of L-function of Legendre symbol modulo 17 is 1.01608483384284
The value of L-function of Legendre symbol modulo 29 is 0.611766289562307
The value of L-function of Legendre symbol modulo 37 is 0.819292168725432
The value of L-function of Legendre symbol modulo 41 is 1.29909306157510
The value of L-function of Legendre symbol modulo 53 is 0.540024945102558
The value of L-function of Legendre symbol modulo 61 is 0.938310198248835
The value of L-function of Legendre symbol modulo 73 is 1.79463648373515
The value of L-function of Legendre symbol modulo 89 is 1.46444140226402
The value of L-function of Legendre symbol modulo 97 is 1.89349532374748
The value of L-function of Legendre symbol modulo 101 is 0.596668668017519
for m in [5,13,17,29,37,41,53,61,73,89,97,101]: _sum_ = 0 for j in [1..(m-1)/2]: _sum_ = _sum_ + kronecker(j,m)*log(sin(j*pi/m)) print "In a cyclic quartic field containing the square root of", m, ", the asymptotic heuristic probability of the Second Factor being a prime number has a ratio of", numerical_approx(2/((2/sqrt(m))*(abs(_sum_)))) 
        
In a cyclic quartic field containing the square root of 5 , the
asymptotic heuristic probability of the Second Factor being a prime
number has a ratio of 4.64674361903477
In a cyclic quartic field containing the square root of 13 , the
asymptotic heuristic probability of the Second Factor being a prime
number has a ratio of 3.01779567975899
In a cyclic quartic field containing the square root of 17 , the
asymptotic heuristic probability of the Second Factor being a prime
number has a ratio of 1.96833958483170
In a cyclic quartic field containing the square root of 29 , the
asymptotic heuristic probability of the Second Factor being a prime
number has a ratio of 3.26922230617008
In a cyclic quartic field containing the square root of 37 , the
asymptotic heuristic probability of the Second Factor being a prime
number has a ratio of 2.44113159669448
In a cyclic quartic field containing the square root of 41 , the
asymptotic heuristic probability of the Second Factor being a prime
number has a ratio of 1.53953558767767
In a cyclic quartic field containing the square root of 53 , the
asymptotic heuristic probability of the Second Factor being a prime
number has a ratio of 3.70353262036844
In a cyclic quartic field containing the square root of 61 , the
asymptotic heuristic probability of the Second Factor being a prime
number has a ratio of 2.13149127413577
In a cyclic quartic field containing the square root of 73 , the
asymptotic heuristic probability of the Second Factor being a prime
number has a ratio of 1.11443181843569
In a cyclic quartic field containing the square root of 89 , the
asymptotic heuristic probability of the Second Factor being a prime
number has a ratio of 1.36570845163761
In a cyclic quartic field containing the square root of 97 , the
asymptotic heuristic probability of the Second Factor being a prime
number has a ratio of 1.05624765739676
In a cyclic quartic field containing the square root of 101 , the
asymptotic heuristic probability of the Second Factor being a prime
number has a ratio of 3.35194406410708
In a cyclic quartic field containing the square root of 5 , the asymptotic heuristic probability of the Second Factor being a prime number has a ratio of 4.64674361903477
In a cyclic quartic field containing the square root of 13 , the asymptotic heuristic probability of the Second Factor being a prime number has a ratio of 3.01779567975899
In a cyclic quartic field containing the square root of 17 , the asymptotic heuristic probability of the Second Factor being a prime number has a ratio of 1.96833958483170
In a cyclic quartic field containing the square root of 29 , the asymptotic heuristic probability of the Second Factor being a prime number has a ratio of 3.26922230617008
In a cyclic quartic field containing the square root of 37 , the asymptotic heuristic probability of the Second Factor being a prime number has a ratio of 2.44113159669448
In a cyclic quartic field containing the square root of 41 , the asymptotic heuristic probability of the Second Factor being a prime number has a ratio of 1.53953558767767
In a cyclic quartic field containing the square root of 53 , the asymptotic heuristic probability of the Second Factor being a prime number has a ratio of 3.70353262036844
In a cyclic quartic field containing the square root of 61 , the asymptotic heuristic probability of the Second Factor being a prime number has a ratio of 2.13149127413577
In a cyclic quartic field containing the square root of 73 , the asymptotic heuristic probability of the Second Factor being a prime number has a ratio of 1.11443181843569
In a cyclic quartic field containing the square root of 89 , the asymptotic heuristic probability of the Second Factor being a prime number has a ratio of 1.36570845163761
In a cyclic quartic field containing the square root of 97 , the asymptotic heuristic probability of the Second Factor being a prime number has a ratio of 1.05624765739676
In a cyclic quartic field containing the square root of 101 , the asymptotic heuristic probability of the Second Factor being a prime number has a ratio of 3.35194406410708
for m in [5, 13, 17, 29, 37, 41, 53, 61, 73, 89, 97, 101, 109, 113, 137, 149, 157, 173, 181, 193, 197, 229, 233, 241, 257, 277, 281, 293, 313, 317, 349, 353, 373, 389, 397, 401, 409, 421, 433, 449, 457, 461]: _sum_ = 0 for j in [1..(m-1)/2]: _sum_ = _sum_ + kronecker(j,m)*log(sin(j*pi/m)) print "The value of L-function of Legendre symbol modulo",m,"is", numerical_approx((2/sqrt(m))*(abs(_sum_))) 
        
The value of L-function of Legendre symbol modulo 5 is 0.430408940964004
The value of L-function of Legendre symbol modulo 13 is
0.662735391071846
The value of L-function of Legendre symbol modulo 17 is 1.01608483384284
The value of L-function of Legendre symbol modulo 29 is
0.611766289562307
The value of L-function of Legendre symbol modulo 37 is
0.819292168725432
The value of L-function of Legendre symbol modulo 41 is 1.29909306157510
The value of L-function of Legendre symbol modulo 53 is
0.540024945102558
The value of L-function of Legendre symbol modulo 61 is
0.938310198248835
The value of L-function of Legendre symbol modulo 73 is 1.79463648373515
The value of L-function of Legendre symbol modulo 89 is 1.46444140226402
The value of L-function of Legendre symbol modulo 97 is 1.89349532374748
The value of L-function of Legendre symbol modulo 101 is
0.596668668017519
The value of L-function of Legendre symbol modulo 109 is
1.06597159422691
The value of L-function of Legendre symbol modulo 113 is
1.38235170906161
The value of L-function of Legendre symbol modulo 137 is
1.39381341332016
The value of L-function of Legendre symbol modulo 149 is
0.673595833301925
The value of L-function of Legendre symbol modulo 157 is
0.855758910337578
The value of L-function of Legendre symbol modulo 173 is
0.390910841189934
The value of L-function of Legendre symbol modulo 181 is
1.06647233219024
The value of L-function of Legendre symbol modulo 193 is
2.17043401934037
The value of L-function of Legendre symbol modulo 197 is
0.475000888532600
The value of L-function of Legendre symbol modulo 229 is
1.07546851605294
The value of L-function of Legendre symbol modulo 233 is
1.40761515138248
The value of L-function of Legendre symbol modulo 241 is
2.41835638390009
The value of L-function of Legendre symbol modulo 257 is
1.29748495883098
The value of L-function of Legendre symbol modulo 277 is
0.945515178592183
The value of L-function of Legendre symbol modulo 281 is
1.73837686726621
The value of L-function of Legendre symbol modulo 293 is
0.331438394291934
The value of L-function of Legendre symbol modulo 313 is
2.18765235235255
The value of L-function of Legendre symbol modulo 317 is
0.504228049302374
The value of L-function of Legendre symbol modulo 349 is
1.05143274302899
The value of L-function of Legendre symbol modulo 353 is
1.26326462707245
The value of L-function of Legendre symbol modulo 373 is
0.956202226690316
The value of L-function of Legendre symbol modulo 389 is
0.795953046556780
The value of L-function of Legendre symbol modulo 397 is
0.817597726937282
The value of L-function of Legendre symbol modulo 401 is
1.84245030922576
The value of L-function of Legendre symbol modulo 409 is
2.58450736883987
The value of L-function of Legendre symbol modulo 421 is
1.26771733328140
The value of L-function of Legendre symbol modulo 433 is
2.24855924140863
The value of L-function of Legendre symbol modulo 449 is
1.86439354511048
The value of L-function of Legendre symbol modulo 457 is
2.38525593186943
The value of L-function of Legendre symbol modulo 461 is
0.549571928481802
The value of L-function of Legendre symbol modulo 5 is 0.430408940964004
The value of L-function of Legendre symbol modulo 13 is 0.662735391071846
The value of L-function of Legendre symbol modulo 17 is 1.01608483384284
The value of L-function of Legendre symbol modulo 29 is 0.611766289562307
The value of L-function of Legendre symbol modulo 37 is 0.819292168725432
The value of L-function of Legendre symbol modulo 41 is 1.29909306157510
The value of L-function of Legendre symbol modulo 53 is 0.540024945102558
The value of L-function of Legendre symbol modulo 61 is 0.938310198248835
The value of L-function of Legendre symbol modulo 73 is 1.79463648373515
The value of L-function of Legendre symbol modulo 89 is 1.46444140226402
The value of L-function of Legendre symbol modulo 97 is 1.89349532374748
The value of L-function of Legendre symbol modulo 101 is 0.596668668017519
The value of L-function of Legendre symbol modulo 109 is 1.06597159422691
The value of L-function of Legendre symbol modulo 113 is 1.38235170906161
The value of L-function of Legendre symbol modulo 137 is 1.39381341332016
The value of L-function of Legendre symbol modulo 149 is 0.673595833301925
The value of L-function of Legendre symbol modulo 157 is 0.855758910337578
The value of L-function of Legendre symbol modulo 173 is 0.390910841189934
The value of L-function of Legendre symbol modulo 181 is 1.06647233219024
The value of L-function of Legendre symbol modulo 193 is 2.17043401934037
The value of L-function of Legendre symbol modulo 197 is 0.475000888532600
The value of L-function of Legendre symbol modulo 229 is 1.07546851605294
The value of L-function of Legendre symbol modulo 233 is 1.40761515138248
The value of L-function of Legendre symbol modulo 241 is 2.41835638390009
The value of L-function of Legendre symbol modulo 257 is 1.29748495883098
The value of L-function of Legendre symbol modulo 277 is 0.945515178592183
The value of L-function of Legendre symbol modulo 281 is 1.73837686726621
The value of L-function of Legendre symbol modulo 293 is 0.331438394291934
The value of L-function of Legendre symbol modulo 313 is 2.18765235235255
The value of L-function of Legendre symbol modulo 317 is 0.504228049302374
The value of L-function of Legendre symbol modulo 349 is 1.05143274302899
The value of L-function of Legendre symbol modulo 353 is 1.26326462707245
The value of L-function of Legendre symbol modulo 373 is 0.956202226690316
The value of L-function of Legendre symbol modulo 389 is 0.795953046556780
The value of L-function of Legendre symbol modulo 397 is 0.817597726937282
The value of L-function of Legendre symbol modulo 401 is 1.84245030922576
The value of L-function of Legendre symbol modulo 409 is 2.58450736883987
The value of L-function of Legendre symbol modulo 421 is 1.26771733328140
The value of L-function of Legendre symbol modulo 433 is 2.24855924140863
The value of L-function of Legendre symbol modulo 449 is 1.86439354511048
The value of L-function of Legendre symbol modulo 457 is 2.38525593186943
The value of L-function of Legendre symbol modulo 461 is 0.549571928481802