L(s) = 1 | + 2.31·2-s − 3-s + 3.35·4-s + 0.113·5-s − 2.31·6-s − 0.994·7-s + 3.13·8-s + 9-s + 0.262·10-s + 0.849·11-s − 3.35·12-s + 13-s − 2.30·14-s − 0.113·15-s + 0.541·16-s − 5.93·17-s + 2.31·18-s − 7.00·19-s + 0.380·20-s + 0.994·21-s + 1.96·22-s − 7.13·23-s − 3.13·24-s − 4.98·25-s + 2.31·26-s − 27-s − 3.33·28-s + ⋯ |
L(s) = 1 | + 1.63·2-s − 0.577·3-s + 1.67·4-s + 0.0507·5-s − 0.944·6-s − 0.375·7-s + 1.10·8-s + 0.333·9-s + 0.0829·10-s + 0.256·11-s − 0.968·12-s + 0.277·13-s − 0.615·14-s − 0.0292·15-s + 0.135·16-s − 1.44·17-s + 0.545·18-s − 1.60·19-s + 0.0850·20-s + 0.217·21-s + 0.419·22-s − 1.48·23-s − 0.639·24-s − 0.997·25-s + 0.453·26-s − 0.192·27-s − 0.630·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 - T \) |
good | 2 | \( 1 - 2.31T + 2T^{2} \) |
| 5 | \( 1 - 0.113T + 5T^{2} \) |
| 7 | \( 1 + 0.994T + 7T^{2} \) |
| 11 | \( 1 - 0.849T + 11T^{2} \) |
| 17 | \( 1 + 5.93T + 17T^{2} \) |
| 19 | \( 1 + 7.00T + 19T^{2} \) |
| 23 | \( 1 + 7.13T + 23T^{2} \) |
| 29 | \( 1 - 0.403T + 29T^{2} \) |
| 31 | \( 1 - 9.98T + 31T^{2} \) |
| 37 | \( 1 - 3.98T + 37T^{2} \) |
| 41 | \( 1 - 5.57T + 41T^{2} \) |
| 43 | \( 1 + 8.50T + 43T^{2} \) |
| 47 | \( 1 + 5.31T + 47T^{2} \) |
| 53 | \( 1 - 0.140T + 53T^{2} \) |
| 59 | \( 1 + 9.09T + 59T^{2} \) |
| 61 | \( 1 - 12.9T + 61T^{2} \) |
| 67 | \( 1 - 11.0T + 67T^{2} \) |
| 71 | \( 1 - 6.28T + 71T^{2} \) |
| 73 | \( 1 + 4.06T + 73T^{2} \) |
| 79 | \( 1 + 11.0T + 79T^{2} \) |
| 83 | \( 1 + 17.6T + 83T^{2} \) |
| 89 | \( 1 - 14.2T + 89T^{2} \) |
| 97 | \( 1 - 7.18T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.945739327244431914701064552206, −6.71985752641356744511954778640, −6.37383950499609933270681093701, −5.97810931091505054841407230441, −4.91173984098875871610791044096, −4.27733641143663333722173810911, −3.82317687013051646936833803719, −2.61670257158007618581994694865, −1.87920350451578293176215547907, 0,
1.87920350451578293176215547907, 2.61670257158007618581994694865, 3.82317687013051646936833803719, 4.27733641143663333722173810911, 4.91173984098875871610791044096, 5.97810931091505054841407230441, 6.37383950499609933270681093701, 6.71985752641356744511954778640, 7.945739327244431914701064552206