| L(s) = 1 | + 2-s + 4-s − 5-s + 7-s + 8-s − 3·9-s − 10-s − 4·13-s + 14-s + 16-s + 4·17-s − 3·18-s + 4·19-s − 20-s − 2·23-s + 25-s − 4·26-s + 28-s − 4·29-s + 2·31-s + 32-s + 4·34-s − 35-s − 3·36-s + 6·37-s + 4·38-s − 40-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.377·7-s + 0.353·8-s − 9-s − 0.316·10-s − 1.10·13-s + 0.267·14-s + 1/4·16-s + 0.970·17-s − 0.707·18-s + 0.917·19-s − 0.223·20-s − 0.417·23-s + 1/5·25-s − 0.784·26-s + 0.188·28-s − 0.742·29-s + 0.359·31-s + 0.176·32-s + 0.685·34-s − 0.169·35-s − 1/2·36-s + 0.986·37-s + 0.648·38-s − 0.158·40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 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 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−7.45688085246516248788368699736, −6.76201743230045416715420891057, −5.85566131979429831279311805131, −5.28966141907234493480692727632, −4.77957113335695135707337425889, −3.82796289913915337201562850725, −3.13320002667192168000941278533, −2.47902468005488213848436318045, −1.37060983363168069560806306197, 0,
1.37060983363168069560806306197, 2.47902468005488213848436318045, 3.13320002667192168000941278533, 3.82796289913915337201562850725, 4.77957113335695135707337425889, 5.28966141907234493480692727632, 5.85566131979429831279311805131, 6.76201743230045416715420891057, 7.45688085246516248788368699736
