| L(s) = 1 | + 2-s + 4-s − 5-s + 4.17·7-s + 8-s − 10-s − 5.69·11-s + 2.73·13-s + 4.17·14-s + 16-s − 3.13·17-s + 0.468·19-s − 20-s − 5.69·22-s − 6.26·23-s + 25-s + 2.73·26-s + 4.17·28-s − 9.71·29-s − 6.42·31-s + 32-s − 3.13·34-s − 4.17·35-s − 5.77·37-s + 0.468·38-s − 40-s − 3.69·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.447·5-s + 1.57·7-s + 0.353·8-s − 0.316·10-s − 1.71·11-s + 0.758·13-s + 1.11·14-s + 0.250·16-s − 0.760·17-s + 0.107·19-s − 0.223·20-s − 1.21·22-s − 1.30·23-s + 0.200·25-s + 0.536·26-s + 0.789·28-s − 1.80·29-s − 1.15·31-s + 0.176·32-s − 0.537·34-s − 0.706·35-s − 0.949·37-s + 0.0759·38-s − 0.158·40-s − 0.576·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 67 | \( 1 + T \) |
| good | 7 | \( 1 - 4.17T + 7T^{2} \) |
| 11 | \( 1 + 5.69T + 11T^{2} \) |
| 13 | \( 1 - 2.73T + 13T^{2} \) |
| 17 | \( 1 + 3.13T + 17T^{2} \) |
| 19 | \( 1 - 0.468T + 19T^{2} \) |
| 23 | \( 1 + 6.26T + 23T^{2} \) |
| 29 | \( 1 + 9.71T + 29T^{2} \) |
| 31 | \( 1 + 6.42T + 31T^{2} \) |
| 37 | \( 1 + 5.77T + 37T^{2} \) |
| 41 | \( 1 + 3.69T + 41T^{2} \) |
| 43 | \( 1 + 8.82T + 43T^{2} \) |
| 47 | \( 1 - 0.179T + 47T^{2} \) |
| 53 | \( 1 - 11.9T + 53T^{2} \) |
| 59 | \( 1 - 0.912T + 59T^{2} \) |
| 61 | \( 1 + 5.42T + 61T^{2} \) |
| 71 | \( 1 - 9.55T + 71T^{2} \) |
| 73 | \( 1 + 15.5T + 73T^{2} \) |
| 79 | \( 1 + 4.02T + 79T^{2} \) |
| 83 | \( 1 + 0.395T + 83T^{2} \) |
| 89 | \( 1 - 5.38T + 89T^{2} \) |
| 97 | \( 1 + 3.36T + 97T^{2} \) |
| 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
−7.52592469893050465951462496399, −7.31233477254678969415574752458, −6.05468705370208624519778952124, −5.37622448810879362641348870833, −4.93307902486004658738882140560, −4.08851621588946779067740350430, −3.42225333196731920798182205432, −2.21693161842788102647846750022, −1.70664271087700849604238305184, 0,
1.70664271087700849604238305184, 2.21693161842788102647846750022, 3.42225333196731920798182205432, 4.08851621588946779067740350430, 4.93307902486004658738882140560, 5.37622448810879362641348870833, 6.05468705370208624519778952124, 7.31233477254678969415574752458, 7.52592469893050465951462496399
