| L(s) = 1 | + 2-s + 4-s − 0.343·5-s − 2.27·7-s + 8-s − 0.343·10-s + 3.94·11-s + 5.39·13-s − 2.27·14-s + 16-s − 7.63·17-s − 4.55·19-s − 0.343·20-s + 3.94·22-s − 4.88·25-s + 5.39·26-s − 2.27·28-s + 2.21·29-s + 2.04·31-s + 32-s − 7.63·34-s + 0.781·35-s + 7.47·37-s − 4.55·38-s − 0.343·40-s + 1.09·41-s + 2.22·43-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.153·5-s − 0.860·7-s + 0.353·8-s − 0.108·10-s + 1.18·11-s + 1.49·13-s − 0.608·14-s + 0.250·16-s − 1.85·17-s − 1.04·19-s − 0.0768·20-s + 0.840·22-s − 0.976·25-s + 1.05·26-s − 0.430·28-s + 0.411·29-s + 0.366·31-s + 0.176·32-s − 1.30·34-s + 0.132·35-s + 1.22·37-s − 0.738·38-s − 0.0543·40-s + 0.170·41-s + 0.339·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9522 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9522 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.039482434\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.039482434\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 23 | \( 1 \) |
| good | 5 | \( 1 + 0.343T + 5T^{2} \) |
| 7 | \( 1 + 2.27T + 7T^{2} \) |
| 11 | \( 1 - 3.94T + 11T^{2} \) |
| 13 | \( 1 - 5.39T + 13T^{2} \) |
| 17 | \( 1 + 7.63T + 17T^{2} \) |
| 19 | \( 1 + 4.55T + 19T^{2} \) |
| 29 | \( 1 - 2.21T + 29T^{2} \) |
| 31 | \( 1 - 2.04T + 31T^{2} \) |
| 37 | \( 1 - 7.47T + 37T^{2} \) |
| 41 | \( 1 - 1.09T + 41T^{2} \) |
| 43 | \( 1 - 2.22T + 43T^{2} \) |
| 47 | \( 1 - 13.2T + 47T^{2} \) |
| 53 | \( 1 + 5.62T + 53T^{2} \) |
| 59 | \( 1 - 8.39T + 59T^{2} \) |
| 61 | \( 1 - 4.48T + 61T^{2} \) |
| 67 | \( 1 - 3.17T + 67T^{2} \) |
| 71 | \( 1 - 3.50T + 71T^{2} \) |
| 73 | \( 1 + 12.0T + 73T^{2} \) |
| 79 | \( 1 - 7.52T + 79T^{2} \) |
| 83 | \( 1 - 5.31T + 83T^{2} \) |
| 89 | \( 1 - 12.5T + 89T^{2} \) |
| 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
−7.53436190283140719148574819387, −6.61615704381637129468272508760, −6.34018993407310667986862104312, −5.94755250795105911484994871163, −4.72422376085035334152555373842, −3.92813452733498601054380129691, −3.85122617755817036427730418607, −2.67900636088808627179721838892, −1.91590632786900661234897657033, −0.74334510141791524606529970884,
0.74334510141791524606529970884, 1.91590632786900661234897657033, 2.67900636088808627179721838892, 3.85122617755817036427730418607, 3.92813452733498601054380129691, 4.72422376085035334152555373842, 5.94755250795105911484994871163, 6.34018993407310667986862104312, 6.61615704381637129468272508760, 7.53436190283140719148574819387
