| L(s) = 1 | + 128·2-s + 1.63e4·4-s − 1.34e5·5-s − 8.23e5·7-s + 2.09e6·8-s − 1.72e7·10-s + 6.80e7·11-s + 1.71e7·13-s − 1.05e8·14-s + 2.68e8·16-s + 1.17e9·17-s − 7.59e9·19-s − 2.21e9·20-s + 8.71e9·22-s + 1.19e9·23-s − 1.22e10·25-s + 2.19e9·26-s − 1.34e10·28-s + 1.13e11·29-s − 7.73e10·31-s + 3.43e10·32-s + 1.50e11·34-s + 1.11e11·35-s + 6.05e11·37-s − 9.72e11·38-s − 2.83e11·40-s − 7.06e11·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.772·5-s − 0.377·7-s + 0.353·8-s − 0.546·10-s + 1.05·11-s + 0.0757·13-s − 0.267·14-s + 0.250·16-s + 0.695·17-s − 1.95·19-s − 0.386·20-s + 0.744·22-s + 0.0733·23-s − 0.402·25-s + 0.0535·26-s − 0.188·28-s + 1.22·29-s − 0.505·31-s + 0.176·32-s + 0.491·34-s + 0.292·35-s + 1.04·37-s − 1.37·38-s − 0.273·40-s − 0.566·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(8)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{17}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 128T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + 8.23e5T \) |
| good | 5 | \( 1 + 1.34e5T + 3.05e10T^{2} \) |
| 11 | \( 1 - 6.80e7T + 4.17e15T^{2} \) |
| 13 | \( 1 - 1.71e7T + 5.11e16T^{2} \) |
| 17 | \( 1 - 1.17e9T + 2.86e18T^{2} \) |
| 19 | \( 1 + 7.59e9T + 1.51e19T^{2} \) |
| 23 | \( 1 - 1.19e9T + 2.66e20T^{2} \) |
| 29 | \( 1 - 1.13e11T + 8.62e21T^{2} \) |
| 31 | \( 1 + 7.73e10T + 2.34e22T^{2} \) |
| 37 | \( 1 - 6.05e11T + 3.33e23T^{2} \) |
| 41 | \( 1 + 7.06e11T + 1.55e24T^{2} \) |
| 43 | \( 1 - 2.30e12T + 3.17e24T^{2} \) |
| 47 | \( 1 - 1.25e12T + 1.20e25T^{2} \) |
| 53 | \( 1 + 6.34e11T + 7.31e25T^{2} \) |
| 59 | \( 1 + 2.49e13T + 3.65e26T^{2} \) |
| 61 | \( 1 - 9.46e12T + 6.02e26T^{2} \) |
| 67 | \( 1 + 1.38e13T + 2.46e27T^{2} \) |
| 71 | \( 1 + 1.42e13T + 5.87e27T^{2} \) |
| 73 | \( 1 - 1.61e13T + 8.90e27T^{2} \) |
| 79 | \( 1 + 3.03e14T + 2.91e28T^{2} \) |
| 83 | \( 1 + 4.80e14T + 6.11e28T^{2} \) |
| 89 | \( 1 - 5.75e14T + 1.74e29T^{2} \) |
| 97 | \( 1 - 1.59e14T + 6.33e29T^{2} \) |
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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
−10.27977260357132086130242417866, −8.969654618114364133995735996417, −7.88360414073055188647058265456, −6.74390297981812866454653427589, −5.93105267404337869993219696606, −4.41659606821310619128407566079, −3.82780582038470009401611119284, −2.64592570946543735598235119068, −1.29687933839997756636607192920, 0,
1.29687933839997756636607192920, 2.64592570946543735598235119068, 3.82780582038470009401611119284, 4.41659606821310619128407566079, 5.93105267404337869993219696606, 6.74390297981812866454653427589, 7.88360414073055188647058265456, 8.969654618114364133995735996417, 10.27977260357132086130242417866
