L(s) = 1 | + 1.77·2-s − 3-s + 1.16·4-s − 3.52·5-s − 1.77·6-s + 0.192·7-s − 1.48·8-s + 9-s − 6.26·10-s + 3.74·11-s − 1.16·12-s + 2.28·13-s + 0.343·14-s + 3.52·15-s − 4.97·16-s + 17-s + 1.77·18-s − 3.71·19-s − 4.10·20-s − 0.192·21-s + 6.65·22-s − 3.43·23-s + 1.48·24-s + 7.41·25-s + 4.07·26-s − 27-s + 0.224·28-s + ⋯ |
L(s) = 1 | + 1.25·2-s − 0.577·3-s + 0.583·4-s − 1.57·5-s − 0.726·6-s + 0.0728·7-s − 0.524·8-s + 0.333·9-s − 1.98·10-s + 1.12·11-s − 0.336·12-s + 0.634·13-s + 0.0916·14-s + 0.909·15-s − 1.24·16-s + 0.242·17-s + 0.419·18-s − 0.851·19-s − 0.918·20-s − 0.0420·21-s + 1.41·22-s − 0.715·23-s + 0.302·24-s + 1.48·25-s + 0.798·26-s − 0.192·27-s + 0.0424·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 157 | \( 1 - T \) |
good | 2 | \( 1 - 1.77T + 2T^{2} \) |
| 5 | \( 1 + 3.52T + 5T^{2} \) |
| 7 | \( 1 - 0.192T + 7T^{2} \) |
| 11 | \( 1 - 3.74T + 11T^{2} \) |
| 13 | \( 1 - 2.28T + 13T^{2} \) |
| 19 | \( 1 + 3.71T + 19T^{2} \) |
| 23 | \( 1 + 3.43T + 23T^{2} \) |
| 29 | \( 1 - 0.184T + 29T^{2} \) |
| 31 | \( 1 - 5.32T + 31T^{2} \) |
| 37 | \( 1 + 1.20T + 37T^{2} \) |
| 41 | \( 1 - 4.39T + 41T^{2} \) |
| 43 | \( 1 - 4.19T + 43T^{2} \) |
| 47 | \( 1 - 6.34T + 47T^{2} \) |
| 53 | \( 1 - 6.09T + 53T^{2} \) |
| 59 | \( 1 + 11.4T + 59T^{2} \) |
| 61 | \( 1 - 3.25T + 61T^{2} \) |
| 67 | \( 1 - 5.23T + 67T^{2} \) |
| 71 | \( 1 - 5.80T + 71T^{2} \) |
| 73 | \( 1 + 9.41T + 73T^{2} \) |
| 79 | \( 1 + 5.46T + 79T^{2} \) |
| 83 | \( 1 - 1.88T + 83T^{2} \) |
| 89 | \( 1 - 11.1T + 89T^{2} \) |
| 97 | \( 1 - 7.34T + 97T^{2} \) |
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
−7.32036789336719695084157964927, −6.45765282667030651949625598843, −6.18837774871041870611154848476, −5.23375809468295290359298744661, −4.42912444865832689634471564890, −4.01766280056443743639946899153, −3.61072378022462353902724712508, −2.57613701259981541053683695582, −1.15964425480926427787740591494, 0,
1.15964425480926427787740591494, 2.57613701259981541053683695582, 3.61072378022462353902724712508, 4.01766280056443743639946899153, 4.42912444865832689634471564890, 5.23375809468295290359298744661, 6.18837774871041870611154848476, 6.45765282667030651949625598843, 7.32036789336719695084157964927