| L(s) = 1 | − 44.2·2-s + 729·3-s − 6.23e3·4-s + 1.48e4·5-s − 3.22e4·6-s + 4.73e5·7-s + 6.38e5·8-s + 5.31e5·9-s − 6.58e5·10-s − 5.62e6·11-s − 4.54e6·12-s − 1.37e7·13-s − 2.09e7·14-s + 1.08e7·15-s + 2.28e7·16-s + 1.90e8·17-s − 2.34e7·18-s − 2.39e8·19-s − 9.29e7·20-s + 3.45e8·21-s + 2.48e8·22-s + 8.48e8·23-s + 4.65e8·24-s − 9.98e8·25-s + 6.07e8·26-s + 3.87e8·27-s − 2.95e9·28-s + ⋯ |
| L(s) = 1 | − 0.488·2-s + 0.577·3-s − 0.761·4-s + 0.426·5-s − 0.282·6-s + 1.52·7-s + 0.860·8-s + 0.333·9-s − 0.208·10-s − 0.957·11-s − 0.439·12-s − 0.790·13-s − 0.743·14-s + 0.246·15-s + 0.340·16-s + 1.91·17-s − 0.162·18-s − 1.16·19-s − 0.324·20-s + 0.878·21-s + 0.467·22-s + 1.19·23-s + 0.496·24-s − 0.818·25-s + 0.385·26-s + 0.192·27-s − 1.15·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 729T \) |
| 59 | \( 1 - 4.21e10T \) |
| good | 2 | \( 1 + 44.2T + 8.19e3T^{2} \) |
| 5 | \( 1 - 1.48e4T + 1.22e9T^{2} \) |
| 7 | \( 1 - 4.73e5T + 9.68e10T^{2} \) |
| 11 | \( 1 + 5.62e6T + 3.45e13T^{2} \) |
| 13 | \( 1 + 1.37e7T + 3.02e14T^{2} \) |
| 17 | \( 1 - 1.90e8T + 9.90e15T^{2} \) |
| 19 | \( 1 + 2.39e8T + 4.20e16T^{2} \) |
| 23 | \( 1 - 8.48e8T + 5.04e17T^{2} \) |
| 29 | \( 1 + 2.26e9T + 1.02e19T^{2} \) |
| 31 | \( 1 - 6.87e8T + 2.44e19T^{2} \) |
| 37 | \( 1 + 1.85e10T + 2.43e20T^{2} \) |
| 41 | \( 1 + 2.65e10T + 9.25e20T^{2} \) |
| 43 | \( 1 + 8.00e10T + 1.71e21T^{2} \) |
| 47 | \( 1 + 1.28e11T + 5.46e21T^{2} \) |
| 53 | \( 1 - 2.46e11T + 2.60e22T^{2} \) |
| 61 | \( 1 - 6.06e11T + 1.61e23T^{2} \) |
| 67 | \( 1 - 1.30e11T + 5.48e23T^{2} \) |
| 71 | \( 1 + 1.05e12T + 1.16e24T^{2} \) |
| 73 | \( 1 + 1.45e12T + 1.67e24T^{2} \) |
| 79 | \( 1 - 9.47e10T + 4.66e24T^{2} \) |
| 83 | \( 1 - 4.49e12T + 8.87e24T^{2} \) |
| 89 | \( 1 - 4.97e11T + 2.19e25T^{2} \) |
| 97 | \( 1 + 1.63e12T + 6.73e25T^{2} \) |
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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
−9.923943365521574502936251444766, −8.649653181404452606741608889820, −8.084299906251712106630025922080, −7.30528934689016606231398548686, −5.31306196330812773647539240917, −4.89530973486165262183017417365, −3.51141720442754803145178748802, −2.08272454709999862387050883480, −1.31662365291902374588883195624, 0,
1.31662365291902374588883195624, 2.08272454709999862387050883480, 3.51141720442754803145178748802, 4.89530973486165262183017417365, 5.31306196330812773647539240917, 7.30528934689016606231398548686, 8.084299906251712106630025922080, 8.649653181404452606741608889820, 9.923943365521574502936251444766
