| L(s) = 1 | + 24.5·2-s + 49.9·3-s + 90.0·4-s − 1.81e3·5-s + 1.22e3·6-s + 8.70e3·7-s − 1.03e4·8-s − 1.71e4·9-s − 4.45e4·10-s + 8.29e4·11-s + 4.50e3·12-s + 2.13e5·14-s − 9.07e4·15-s − 3.00e5·16-s + 3.74e5·17-s − 4.21e5·18-s + 3.61e5·19-s − 1.63e5·20-s + 4.35e5·21-s + 2.03e6·22-s − 2.31e6·23-s − 5.17e5·24-s + 1.34e6·25-s − 1.84e6·27-s + 7.83e5·28-s − 6.49e5·29-s − 2.22e6·30-s + ⋯ |
| L(s) = 1 | + 1.08·2-s + 0.356·3-s + 0.175·4-s − 1.29·5-s + 0.386·6-s + 1.37·7-s − 0.893·8-s − 0.873·9-s − 1.40·10-s + 1.70·11-s + 0.0626·12-s + 1.48·14-s − 0.462·15-s − 1.14·16-s + 1.08·17-s − 0.946·18-s + 0.636·19-s − 0.228·20-s + 0.488·21-s + 1.85·22-s − 1.72·23-s − 0.318·24-s + 0.686·25-s − 0.667·27-s + 0.240·28-s − 0.170·29-s − 0.501·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 \) |
| good | 2 | \( 1 - 24.5T + 512T^{2} \) |
| 3 | \( 1 - 49.9T + 1.96e4T^{2} \) |
| 5 | \( 1 + 1.81e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 8.70e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 8.29e4T + 2.35e9T^{2} \) |
| 17 | \( 1 - 3.74e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 3.61e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 2.31e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 6.49e5T + 1.45e13T^{2} \) |
| 31 | \( 1 + 4.32e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.21e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 2.59e6T + 3.27e14T^{2} \) |
| 43 | \( 1 - 3.08e6T + 5.02e14T^{2} \) |
| 47 | \( 1 + 2.60e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 1.01e8T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.37e8T + 8.66e15T^{2} \) |
| 61 | \( 1 + 4.29e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 5.54e7T + 2.72e16T^{2} \) |
| 71 | \( 1 - 1.43e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 3.37e7T + 5.88e16T^{2} \) |
| 79 | \( 1 + 2.66e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 9.89e7T + 1.86e17T^{2} \) |
| 89 | \( 1 - 5.08e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 4.53e8T + 7.60e17T^{2} \) |
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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
−11.19753665237841359338559398181, −9.353254039428723319871992824324, −8.364498290874469947983191674449, −7.63260386179063267935671012699, −6.08646671200236855495067991256, −4.96134839808983785881099866051, −3.93521097688111131934080267540, −3.35660515892335501071083341130, −1.58747777393703079772071775130, 0,
1.58747777393703079772071775130, 3.35660515892335501071083341130, 3.93521097688111131934080267540, 4.96134839808983785881099866051, 6.08646671200236855495067991256, 7.63260386179063267935671012699, 8.364498290874469947983191674449, 9.353254039428723319871992824324, 11.19753665237841359338559398181
