| L(s) = 1 | − 2·5-s − 3·9-s − 4·11-s + 2·13-s − 6·17-s − 23-s − 25-s − 6·29-s − 4·31-s + 6·37-s − 10·41-s + 4·43-s + 6·45-s − 7·49-s − 2·53-s + 8·55-s − 8·59-s + 6·61-s − 4·65-s + 12·67-s − 12·71-s + 10·73-s + 8·79-s + 9·81-s + 12·83-s + 12·85-s + 2·89-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 9-s − 1.20·11-s + 0.554·13-s − 1.45·17-s − 0.208·23-s − 1/5·25-s − 1.11·29-s − 0.718·31-s + 0.986·37-s − 1.56·41-s + 0.609·43-s + 0.894·45-s − 49-s − 0.274·53-s + 1.07·55-s − 1.04·59-s + 0.768·61-s − 0.496·65-s + 1.46·67-s − 1.42·71-s + 1.17·73-s + 0.900·79-s + 81-s + 1.31·83-s + 1.30·85-s + 0.211·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 66424 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 66424 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−14.61311030653740, −13.83643229498129, −13.40621872867574, −13.00082792424812, −12.51301639640385, −11.72708496653402, −11.43533454054111, −10.94824942910186, −10.68142113712284, −9.856943606944404, −9.258046884686869, −8.718004672732549, −8.283614901968703, −7.743953349172700, −7.426632598530276, −6.534036736032466, −6.149427775368817, −5.441717410481120, −4.934494014253564, −4.338410629080401, −3.547814291513493, −3.272888957948160, −2.302796604451501, −1.953060080059687, −0.5895519527175478, 0,
0.5895519527175478, 1.953060080059687, 2.302796604451501, 3.272888957948160, 3.547814291513493, 4.338410629080401, 4.934494014253564, 5.441717410481120, 6.149427775368817, 6.534036736032466, 7.426632598530276, 7.743953349172700, 8.283614901968703, 8.718004672732549, 9.258046884686869, 9.856943606944404, 10.68142113712284, 10.94824942910186, 11.43533454054111, 11.72708496653402, 12.51301639640385, 13.00082792424812, 13.40621872867574, 13.83643229498129, 14.61311030653740
