| L(s) = 1 | − 62.2·2-s + 1.83e3·4-s − 1.37e3·5-s + 3.30e4·7-s + 1.35e4·8-s + 8.57e4·10-s + 8.00e5·11-s + 1.48e6·13-s − 2.05e6·14-s − 4.59e6·16-s + 1.13e7·17-s − 9.78e6·19-s − 2.52e6·20-s − 4.98e7·22-s − 5.46e5·23-s − 4.69e7·25-s − 9.25e7·26-s + 6.04e7·28-s − 2.05e7·29-s + 4.17e6·31-s + 2.58e8·32-s − 7.03e8·34-s − 4.55e7·35-s + 3.50e8·37-s + 6.09e8·38-s − 1.86e7·40-s − 7.34e8·41-s + ⋯ |
| L(s) = 1 | − 1.37·2-s + 0.893·4-s − 0.197·5-s + 0.743·7-s + 0.146·8-s + 0.271·10-s + 1.49·11-s + 1.10·13-s − 1.02·14-s − 1.09·16-s + 1.93·17-s − 0.906·19-s − 0.176·20-s − 2.06·22-s − 0.0176·23-s − 0.961·25-s − 1.52·26-s + 0.664·28-s − 0.185·29-s + 0.0262·31-s + 1.36·32-s − 2.65·34-s − 0.146·35-s + 0.831·37-s + 1.24·38-s − 0.0288·40-s − 0.989·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 29 | \( 1 + 2.05e7T \) |
| good | 2 | \( 1 + 62.2T + 2.04e3T^{2} \) |
| 5 | \( 1 + 1.37e3T + 4.88e7T^{2} \) |
| 7 | \( 1 - 3.30e4T + 1.97e9T^{2} \) |
| 11 | \( 1 - 8.00e5T + 2.85e11T^{2} \) |
| 13 | \( 1 - 1.48e6T + 1.79e12T^{2} \) |
| 17 | \( 1 - 1.13e7T + 3.42e13T^{2} \) |
| 19 | \( 1 + 9.78e6T + 1.16e14T^{2} \) |
| 23 | \( 1 + 5.46e5T + 9.52e14T^{2} \) |
| 31 | \( 1 - 4.17e6T + 2.54e16T^{2} \) |
| 37 | \( 1 - 3.50e8T + 1.77e17T^{2} \) |
| 41 | \( 1 + 7.34e8T + 5.50e17T^{2} \) |
| 43 | \( 1 + 1.74e9T + 9.29e17T^{2} \) |
| 47 | \( 1 + 1.70e9T + 2.47e18T^{2} \) |
| 53 | \( 1 + 2.30e9T + 9.26e18T^{2} \) |
| 59 | \( 1 - 2.84e8T + 3.01e19T^{2} \) |
| 61 | \( 1 - 5.13e9T + 4.35e19T^{2} \) |
| 67 | \( 1 + 3.85e9T + 1.22e20T^{2} \) |
| 71 | \( 1 + 2.82e10T + 2.31e20T^{2} \) |
| 73 | \( 1 + 1.81e10T + 3.13e20T^{2} \) |
| 79 | \( 1 + 3.94e10T + 7.47e20T^{2} \) |
| 83 | \( 1 + 1.27e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + 4.46e10T + 2.77e21T^{2} \) |
| 97 | \( 1 + 7.40e10T + 7.15e21T^{2} \) |
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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
−9.549022115639588834062206950290, −8.483118281345838786990076686335, −8.077864294354268715744311284788, −6.95341236873167870078893138429, −5.91941422220437263762048100077, −4.45228748799176651076112158963, −3.43215741367399115704106397939, −1.55965762085683838389536612806, −1.33568785201558267920521863121, 0,
1.33568785201558267920521863121, 1.55965762085683838389536612806, 3.43215741367399115704106397939, 4.45228748799176651076112158963, 5.91941422220437263762048100077, 6.95341236873167870078893138429, 8.077864294354268715744311284788, 8.483118281345838786990076686335, 9.549022115639588834062206950290
