| L(s) = 1 | + 2-s − 3-s + 4-s + 2.78·5-s − 6-s − 7-s + 8-s + 9-s + 2.78·10-s + 3.87·11-s − 12-s − 1.77·13-s − 14-s − 2.78·15-s + 16-s − 4.92·17-s + 18-s − 2.92·19-s + 2.78·20-s + 21-s + 3.87·22-s − 7.45·23-s − 24-s + 2.77·25-s − 1.77·26-s − 27-s − 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.24·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 0.333·9-s + 0.881·10-s + 1.16·11-s − 0.288·12-s − 0.491·13-s − 0.267·14-s − 0.719·15-s + 0.250·16-s − 1.19·17-s + 0.235·18-s − 0.671·19-s + 0.623·20-s + 0.218·21-s + 0.825·22-s − 1.55·23-s − 0.204·24-s + 0.554·25-s − 0.347·26-s − 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 191 | \( 1 - T \) |
| good | 5 | \( 1 - 2.78T + 5T^{2} \) |
| 11 | \( 1 - 3.87T + 11T^{2} \) |
| 13 | \( 1 + 1.77T + 13T^{2} \) |
| 17 | \( 1 + 4.92T + 17T^{2} \) |
| 19 | \( 1 + 2.92T + 19T^{2} \) |
| 23 | \( 1 + 7.45T + 23T^{2} \) |
| 29 | \( 1 - 2.32T + 29T^{2} \) |
| 31 | \( 1 + 7.00T + 31T^{2} \) |
| 37 | \( 1 + 6.91T + 37T^{2} \) |
| 41 | \( 1 - 5.49T + 41T^{2} \) |
| 43 | \( 1 + 6.49T + 43T^{2} \) |
| 47 | \( 1 + 3.08T + 47T^{2} \) |
| 53 | \( 1 + 10.1T + 53T^{2} \) |
| 59 | \( 1 + 0.307T + 59T^{2} \) |
| 61 | \( 1 + 5.03T + 61T^{2} \) |
| 67 | \( 1 + 9.27T + 67T^{2} \) |
| 71 | \( 1 + 9.22T + 71T^{2} \) |
| 73 | \( 1 - 8.55T + 73T^{2} \) |
| 79 | \( 1 + 15.1T + 79T^{2} \) |
| 83 | \( 1 - 9.23T + 83T^{2} \) |
| 89 | \( 1 - 0.0377T + 89T^{2} \) |
| 97 | \( 1 + 8.64T + 97T^{2} \) |
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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
−7.05871955306697837753992832817, −6.56389070663513478789271189191, −6.08798458710599123002358219126, −5.55925386856531211460119970876, −4.63473079803911440853064006910, −4.12025640017444834737785137461, −3.16060763301504794547636932089, −2.02304249114675552539908210760, −1.68498759549438841542568875162, 0,
1.68498759549438841542568875162, 2.02304249114675552539908210760, 3.16060763301504794547636932089, 4.12025640017444834737785137461, 4.63473079803911440853064006910, 5.55925386856531211460119970876, 6.08798458710599123002358219126, 6.56389070663513478789271189191, 7.05871955306697837753992832817
