| L(s) = 1 | + 10.6·2-s + 27·3-s − 15.6·4-s + 100.·5-s + 286.·6-s − 1.21e3·7-s − 1.52e3·8-s + 729·9-s + 1.06e3·10-s − 5.14e3·11-s − 421.·12-s − 4.33e3·13-s − 1.28e4·14-s + 2.70e3·15-s − 1.41e4·16-s + 2.80e4·17-s + 7.72e3·18-s − 2.47e4·19-s − 1.56e3·20-s − 3.28e4·21-s − 5.45e4·22-s − 1.21e4·23-s − 4.11e4·24-s − 6.80e4·25-s − 4.59e4·26-s + 1.96e4·27-s + 1.89e4·28-s + ⋯ |
| L(s) = 1 | + 0.937·2-s + 0.577·3-s − 0.122·4-s + 0.358·5-s + 0.540·6-s − 1.34·7-s − 1.05·8-s + 0.333·9-s + 0.335·10-s − 1.16·11-s − 0.0704·12-s − 0.547·13-s − 1.25·14-s + 0.206·15-s − 0.863·16-s + 1.38·17-s + 0.312·18-s − 0.828·19-s − 0.0437·20-s − 0.773·21-s − 1.09·22-s − 0.208·23-s − 0.606·24-s − 0.871·25-s − 0.512·26-s + 0.192·27-s + 0.163·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - 27T \) |
| 23 | \( 1 + 1.21e4T \) |
| good | 2 | \( 1 - 10.6T + 128T^{2} \) |
| 5 | \( 1 - 100.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 1.21e3T + 8.23e5T^{2} \) |
| 11 | \( 1 + 5.14e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 4.33e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 2.80e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 2.47e4T + 8.93e8T^{2} \) |
| 29 | \( 1 - 3.98e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 7.56e3T + 2.75e10T^{2} \) |
| 37 | \( 1 - 2.52e4T + 9.49e10T^{2} \) |
| 41 | \( 1 - 4.92e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 1.01e6T + 2.71e11T^{2} \) |
| 47 | \( 1 + 7.91e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.33e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 6.21e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + 2.33e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 1.54e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 2.85e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 3.85e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 3.15e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 3.03e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 6.07e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.02e7T + 8.07e13T^{2} \) |
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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
−12.97696320131642341747417746333, −12.20885831933307069761754400589, −10.18398998923193257786350314476, −9.468990238784989801643720096397, −7.994298333050552112460109163002, −6.37899551487593732994131802439, −5.20716439568486487789335616429, −3.63718078557932910482970252457, −2.60732269644847474723727868432, 0,
2.60732269644847474723727868432, 3.63718078557932910482970252457, 5.20716439568486487789335616429, 6.37899551487593732994131802439, 7.994298333050552112460109163002, 9.468990238784989801643720096397, 10.18398998923193257786350314476, 12.20885831933307069761754400589, 12.97696320131642341747417746333
