| L(s) = 1 | + 2-s − 3-s + 4-s − 3.75·5-s − 6-s + 7-s + 8-s + 9-s − 3.75·10-s − 1.78·11-s − 12-s − 2.24·13-s + 14-s + 3.75·15-s + 16-s − 5.67·17-s + 18-s + 5.23·19-s − 3.75·20-s − 21-s − 1.78·22-s + 4.05·23-s − 24-s + 9.09·25-s − 2.24·26-s − 27-s + 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.67·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s − 1.18·10-s − 0.538·11-s − 0.288·12-s − 0.622·13-s + 0.267·14-s + 0.969·15-s + 0.250·16-s − 1.37·17-s + 0.235·18-s + 1.20·19-s − 0.839·20-s − 0.218·21-s − 0.380·22-s + 0.846·23-s − 0.204·24-s + 1.81·25-s − 0.440·26-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 | \( 1 + 3.75T + 5T^{2} \) |
| 11 | \( 1 + 1.78T + 11T^{2} \) |
| 13 | \( 1 + 2.24T + 13T^{2} \) |
| 17 | \( 1 + 5.67T + 17T^{2} \) |
| 19 | \( 1 - 5.23T + 19T^{2} \) |
| 23 | \( 1 - 4.05T + 23T^{2} \) |
| 29 | \( 1 - 4.60T + 29T^{2} \) |
| 31 | \( 1 + 0.510T + 31T^{2} \) |
| 37 | \( 1 - 9.39T + 37T^{2} \) |
| 41 | \( 1 + 0.356T + 41T^{2} \) |
| 43 | \( 1 + 0.223T + 43T^{2} \) |
| 47 | \( 1 + 4.51T + 47T^{2} \) |
| 53 | \( 1 + 1.81T + 53T^{2} \) |
| 59 | \( 1 - 5.12T + 59T^{2} \) |
| 61 | \( 1 + 6.96T + 61T^{2} \) |
| 67 | \( 1 + 3.96T + 67T^{2} \) |
| 71 | \( 1 + 8.49T + 71T^{2} \) |
| 73 | \( 1 - 3.07T + 73T^{2} \) |
| 79 | \( 1 + 3.06T + 79T^{2} \) |
| 83 | \( 1 - 6.87T + 83T^{2} \) |
| 89 | \( 1 + 9.23T + 89T^{2} \) |
| 97 | \( 1 + 4.33T + 97T^{2} \) |
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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
−7.37783807487276458832929551791, −6.91376396656654067936631590691, −6.07215550078844033365691903988, −5.06049877745408242648241539367, −4.70929793189166494518523259739, −4.13263834251181590928012977946, −3.20747365870935061203211913270, −2.51952142503264599521512107669, −1.10582699659848682192838844677, 0,
1.10582699659848682192838844677, 2.51952142503264599521512107669, 3.20747365870935061203211913270, 4.13263834251181590928012977946, 4.70929793189166494518523259739, 5.06049877745408242648241539367, 6.07215550078844033365691903988, 6.91376396656654067936631590691, 7.37783807487276458832929551791
