| L(s) = 1 | − 2.21·3-s + 5-s + 1.52·7-s + 1.90·9-s + 2.90·11-s − 1.31·13-s − 2.21·15-s − 7.80·17-s + 19-s − 3.37·21-s + 4.28·23-s + 25-s + 2.42·27-s − 9.18·29-s − 7.80·31-s − 6.42·33-s + 1.52·35-s + 8.16·37-s + 2.90·39-s − 11.0·41-s + 1.09·43-s + 1.90·45-s + 4.28·47-s − 4.67·49-s + 17.2·51-s + 5.54·53-s + 2.90·55-s + ⋯ |
| L(s) = 1 | − 1.27·3-s + 0.447·5-s + 0.576·7-s + 0.634·9-s + 0.875·11-s − 0.363·13-s − 0.571·15-s − 1.89·17-s + 0.229·19-s − 0.737·21-s + 0.892·23-s + 0.200·25-s + 0.467·27-s − 1.70·29-s − 1.40·31-s − 1.11·33-s + 0.257·35-s + 1.34·37-s + 0.464·39-s − 1.72·41-s + 0.167·43-s + 0.283·45-s + 0.624·47-s − 0.667·49-s + 2.42·51-s + 0.761·53-s + 0.391·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 2.21T + 3T^{2} \) |
| 7 | \( 1 - 1.52T + 7T^{2} \) |
| 11 | \( 1 - 2.90T + 11T^{2} \) |
| 13 | \( 1 + 1.31T + 13T^{2} \) |
| 17 | \( 1 + 7.80T + 17T^{2} \) |
| 23 | \( 1 - 4.28T + 23T^{2} \) |
| 29 | \( 1 + 9.18T + 29T^{2} \) |
| 31 | \( 1 + 7.80T + 31T^{2} \) |
| 37 | \( 1 - 8.16T + 37T^{2} \) |
| 41 | \( 1 + 11.0T + 41T^{2} \) |
| 43 | \( 1 - 1.09T + 43T^{2} \) |
| 47 | \( 1 - 4.28T + 47T^{2} \) |
| 53 | \( 1 - 5.54T + 53T^{2} \) |
| 59 | \( 1 + 1.86T + 59T^{2} \) |
| 61 | \( 1 - 0.709T + 61T^{2} \) |
| 67 | \( 1 - 1.45T + 67T^{2} \) |
| 71 | \( 1 - 2.29T + 71T^{2} \) |
| 73 | \( 1 - 14.0T + 73T^{2} \) |
| 79 | \( 1 + 6.13T + 79T^{2} \) |
| 83 | \( 1 + 11.7T + 83T^{2} \) |
| 89 | \( 1 - 3.67T + 89T^{2} \) |
| 97 | \( 1 + 11.2T + 97T^{2} \) |
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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
−8.479287067223291568801457144413, −7.25434860183952808228317534126, −6.78824696016997509612977609771, −6.00825459924150429446555468115, −5.29125501088101962842728215891, −4.67744456902282165995443375809, −3.78459189862710463516581790233, −2.34203169077999906920532045271, −1.38232949222341698401447607469, 0,
1.38232949222341698401447607469, 2.34203169077999906920532045271, 3.78459189862710463516581790233, 4.67744456902282165995443375809, 5.29125501088101962842728215891, 6.00825459924150429446555468115, 6.78824696016997509612977609771, 7.25434860183952808228317534126, 8.479287067223291568801457144413
