L(s) = 1 | − 0.328·3-s − 1.06·5-s + 1.36·7-s − 2.89·9-s + 0.558·11-s − 2.36·13-s + 0.348·15-s + 17-s + 7.37·19-s − 0.448·21-s + 0.317·23-s − 3.87·25-s + 1.93·27-s + 0.244·29-s + 2.63·31-s − 0.183·33-s − 1.44·35-s − 6.34·37-s + 0.777·39-s − 10.2·41-s − 0.829·43-s + 3.07·45-s + 12.5·47-s − 5.13·49-s − 0.328·51-s − 2.84·53-s − 0.593·55-s + ⋯ |
L(s) = 1 | − 0.189·3-s − 0.474·5-s + 0.516·7-s − 0.963·9-s + 0.168·11-s − 0.655·13-s + 0.0901·15-s + 0.242·17-s + 1.69·19-s − 0.0979·21-s + 0.0661·23-s − 0.774·25-s + 0.372·27-s + 0.0453·29-s + 0.473·31-s − 0.0319·33-s − 0.245·35-s − 1.04·37-s + 0.124·39-s − 1.60·41-s − 0.126·43-s + 0.457·45-s + 1.83·47-s − 0.733·49-s − 0.0460·51-s − 0.390·53-s − 0.0800·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 - T \) |
good | 3 | \( 1 + 0.328T + 3T^{2} \) |
| 5 | \( 1 + 1.06T + 5T^{2} \) |
| 7 | \( 1 - 1.36T + 7T^{2} \) |
| 11 | \( 1 - 0.558T + 11T^{2} \) |
| 13 | \( 1 + 2.36T + 13T^{2} \) |
| 19 | \( 1 - 7.37T + 19T^{2} \) |
| 23 | \( 1 - 0.317T + 23T^{2} \) |
| 29 | \( 1 - 0.244T + 29T^{2} \) |
| 31 | \( 1 - 2.63T + 31T^{2} \) |
| 37 | \( 1 + 6.34T + 37T^{2} \) |
| 41 | \( 1 + 10.2T + 41T^{2} \) |
| 43 | \( 1 + 0.829T + 43T^{2} \) |
| 47 | \( 1 - 12.5T + 47T^{2} \) |
| 53 | \( 1 + 2.84T + 53T^{2} \) |
| 61 | \( 1 + 0.307T + 61T^{2} \) |
| 67 | \( 1 + 2.20T + 67T^{2} \) |
| 71 | \( 1 - 4.35T + 71T^{2} \) |
| 73 | \( 1 - 13.0T + 73T^{2} \) |
| 79 | \( 1 + 5.52T + 79T^{2} \) |
| 83 | \( 1 + 5.43T + 83T^{2} \) |
| 89 | \( 1 - 6.94T + 89T^{2} \) |
| 97 | \( 1 - 13.1T + 97T^{2} \) |
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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
−7.56778168431938735598719600415, −6.89748453123137562352011479471, −6.02519473453722552050385253387, −5.24140918272134097294533119325, −4.91833229823245574040243726536, −3.79179268989463746808212336499, −3.18529316343842316132510626105, −2.27465506305342091813919753644, −1.16577496599860703770362407845, 0,
1.16577496599860703770362407845, 2.27465506305342091813919753644, 3.18529316343842316132510626105, 3.79179268989463746808212336499, 4.91833229823245574040243726536, 5.24140918272134097294533119325, 6.02519473453722552050385253387, 6.89748453123137562352011479471, 7.56778168431938735598719600415