| L(s) = 1 | − 2·2-s + 4·4-s + 15.2·5-s − 8·8-s − 30.4·10-s − 2·11-s + 30.4·13-s + 16·16-s − 45.6·17-s − 152.·19-s + 60.9·20-s + 4·22-s − 30·23-s + 107.·25-s − 60.9·26-s − 212·29-s − 213.·31-s − 32·32-s + 91.3·34-s + 246·37-s + 304.·38-s − 121.·40-s − 319.·41-s − 284·43-s − 8·44-s + 60·46-s + 60.9·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.36·5-s − 0.353·8-s − 0.963·10-s − 0.0548·11-s + 0.649·13-s + 0.250·16-s − 0.651·17-s − 1.83·19-s + 0.681·20-s + 0.0387·22-s − 0.271·23-s + 0.856·25-s − 0.459·26-s − 1.35·29-s − 1.23·31-s − 0.176·32-s + 0.460·34-s + 1.09·37-s + 1.30·38-s − 0.481·40-s − 1.21·41-s − 1.00·43-s − 0.0274·44-s + 0.192·46-s + 0.189·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 15.2T + 125T^{2} \) |
| 11 | \( 1 + 2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 30.4T + 2.19e3T^{2} \) |
| 17 | \( 1 + 45.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 152.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 30T + 1.21e4T^{2} \) |
| 29 | \( 1 + 212T + 2.43e4T^{2} \) |
| 31 | \( 1 + 213.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 246T + 5.06e4T^{2} \) |
| 41 | \( 1 + 319.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 284T + 7.95e4T^{2} \) |
| 47 | \( 1 - 60.9T + 1.03e5T^{2} \) |
| 53 | \( 1 + 548T + 1.48e5T^{2} \) |
| 59 | \( 1 + 670.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 517.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 652T + 3.00e5T^{2} \) |
| 71 | \( 1 + 770T + 3.57e5T^{2} \) |
| 73 | \( 1 - 974.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 472T + 4.93e5T^{2} \) |
| 83 | \( 1 - 182.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 715.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 304.T + 9.12e5T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.276940348827369100559630365676, −8.730775018060954274199349119871, −7.76843512357802153775511473467, −6.53122590028163273720299836186, −6.15802044935025213151043510973, −5.08117333641296461362158615117, −3.72517154864632884131968303331, −2.25978069439975492284996951056, −1.66004194316077754044389828864, 0,
1.66004194316077754044389828864, 2.25978069439975492284996951056, 3.72517154864632884131968303331, 5.08117333641296461362158615117, 6.15802044935025213151043510973, 6.53122590028163273720299836186, 7.76843512357802153775511473467, 8.730775018060954274199349119871, 9.276940348827369100559630365676
