| L(s) = 1 | + 2-s + 4-s − 1.40·5-s − 2.13·7-s + 8-s − 1.40·10-s − 2.83·11-s + 5.71·13-s − 2.13·14-s + 16-s + 5.87·17-s − 3.20·19-s − 1.40·20-s − 2.83·22-s − 3.97·23-s − 3.03·25-s + 5.71·26-s − 2.13·28-s + 0.100·29-s − 5.43·31-s + 32-s + 5.87·34-s + 2.99·35-s + 6.63·37-s − 3.20·38-s − 1.40·40-s − 1.02·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.626·5-s − 0.807·7-s + 0.353·8-s − 0.442·10-s − 0.854·11-s + 1.58·13-s − 0.570·14-s + 0.250·16-s + 1.42·17-s − 0.736·19-s − 0.313·20-s − 0.604·22-s − 0.828·23-s − 0.607·25-s + 1.12·26-s − 0.403·28-s + 0.0186·29-s − 0.976·31-s + 0.176·32-s + 1.00·34-s + 0.505·35-s + 1.09·37-s − 0.520·38-s − 0.221·40-s − 0.160·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 149 | \( 1 - T \) |
| good | 5 | \( 1 + 1.40T + 5T^{2} \) |
| 7 | \( 1 + 2.13T + 7T^{2} \) |
| 11 | \( 1 + 2.83T + 11T^{2} \) |
| 13 | \( 1 - 5.71T + 13T^{2} \) |
| 17 | \( 1 - 5.87T + 17T^{2} \) |
| 19 | \( 1 + 3.20T + 19T^{2} \) |
| 23 | \( 1 + 3.97T + 23T^{2} \) |
| 29 | \( 1 - 0.100T + 29T^{2} \) |
| 31 | \( 1 + 5.43T + 31T^{2} \) |
| 37 | \( 1 - 6.63T + 37T^{2} \) |
| 41 | \( 1 + 1.02T + 41T^{2} \) |
| 43 | \( 1 + 3.45T + 43T^{2} \) |
| 47 | \( 1 - 4.46T + 47T^{2} \) |
| 53 | \( 1 - 9.91T + 53T^{2} \) |
| 59 | \( 1 + 9.99T + 59T^{2} \) |
| 61 | \( 1 + 0.434T + 61T^{2} \) |
| 67 | \( 1 + 9.79T + 67T^{2} \) |
| 71 | \( 1 - 11.9T + 71T^{2} \) |
| 73 | \( 1 + 11.4T + 73T^{2} \) |
| 79 | \( 1 + 0.914T + 79T^{2} \) |
| 83 | \( 1 + 9.47T + 83T^{2} \) |
| 89 | \( 1 - 8.06T + 89T^{2} \) |
| 97 | \( 1 - 0.417T + 97T^{2} \) |
| 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
−7.57824697829247888906851473294, −6.64001368267669466598453429911, −5.91113945823784725037475478593, −5.62835294363312121226529497324, −4.51421149520209937249250312586, −3.74750170741284628344053036120, −3.39750194153659680168080021874, −2.46495984259136681361221161002, −1.32770669344170694071414737484, 0,
1.32770669344170694071414737484, 2.46495984259136681361221161002, 3.39750194153659680168080021874, 3.74750170741284628344053036120, 4.51421149520209937249250312586, 5.62835294363312121226529497324, 5.91113945823784725037475478593, 6.64001368267669466598453429911, 7.57824697829247888906851473294
