| L(s) = 1 | + 8·2-s + 36.1·3-s + 64·4-s + 289.·6-s − 343·7-s + 512·8-s − 878.·9-s + 6.31e3·11-s + 2.31e3·12-s − 745.·13-s − 2.74e3·14-s + 4.09e3·16-s − 3.42e4·17-s − 7.02e3·18-s − 4.59e4·19-s − 1.24e4·21-s + 5.05e4·22-s − 5.82e4·23-s + 1.85e4·24-s − 5.96e3·26-s − 1.10e5·27-s − 2.19e4·28-s + 9.85e4·29-s + 3.82e3·31-s + 3.27e4·32-s + 2.28e5·33-s − 2.73e5·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.773·3-s + 0.5·4-s + 0.546·6-s − 0.377·7-s + 0.353·8-s − 0.401·9-s + 1.43·11-s + 0.386·12-s − 0.0940·13-s − 0.267·14-s + 0.250·16-s − 1.68·17-s − 0.284·18-s − 1.53·19-s − 0.292·21-s + 1.01·22-s − 0.998·23-s + 0.273·24-s − 0.0665·26-s − 1.08·27-s − 0.188·28-s + 0.750·29-s + 0.0230·31-s + 0.176·32-s + 1.10·33-s − 1.19·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 8T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 343T \) |
| good | 3 | \( 1 - 36.1T + 2.18e3T^{2} \) |
| 11 | \( 1 - 6.31e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 745.T + 6.27e7T^{2} \) |
| 17 | \( 1 + 3.42e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 4.59e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 5.82e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 9.85e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 3.82e3T + 2.75e10T^{2} \) |
| 37 | \( 1 - 5.06e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 2.05e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 8.72e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 9.38e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 4.83e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 2.72e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 3.30e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 4.13e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 4.30e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 9.08e5T + 1.10e13T^{2} \) |
| 79 | \( 1 + 5.74e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 2.66e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 1.38e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.51e7T + 8.07e13T^{2} \) |
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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
−9.688574063521328648274737526386, −8.846736451371390052297249119306, −8.062731157752663095466753105499, −6.58321889376623222782869900119, −6.24401466675511810785027548691, −4.55209464707953246902161305376, −3.84302256751810834908591536773, −2.68698779079375998633859859241, −1.78059878511026264715885590493, 0,
1.78059878511026264715885590493, 2.68698779079375998633859859241, 3.84302256751810834908591536773, 4.55209464707953246902161305376, 6.24401466675511810785027548691, 6.58321889376623222782869900119, 8.062731157752663095466753105499, 8.846736451371390052297249119306, 9.688574063521328648274737526386
