| L(s) = 1 | + 61.1·2-s + 1.68e3·4-s − 3.36e3·5-s + 1.07e4·7-s − 2.19e4·8-s − 2.05e5·10-s − 2.39e4·11-s + 2.37e6·13-s + 6.59e5·14-s − 4.80e6·16-s − 8.22e6·17-s + 6.96e6·19-s − 5.68e6·20-s − 1.46e6·22-s − 3.92e7·23-s − 3.74e7·25-s + 1.44e8·26-s + 1.82e7·28-s − 1.43e8·29-s − 1.40e8·31-s − 2.48e8·32-s − 5.02e8·34-s − 3.63e7·35-s − 9.06e7·37-s + 4.25e8·38-s + 7.39e7·40-s + 1.06e9·41-s + ⋯ |
| L(s) = 1 | + 1.35·2-s + 0.824·4-s − 0.482·5-s + 0.242·7-s − 0.236·8-s − 0.651·10-s − 0.0447·11-s + 1.77·13-s + 0.327·14-s − 1.14·16-s − 1.40·17-s + 0.645·19-s − 0.397·20-s − 0.0604·22-s − 1.27·23-s − 0.767·25-s + 2.39·26-s + 0.200·28-s − 1.29·29-s − 0.879·31-s − 1.30·32-s − 1.89·34-s − 0.117·35-s − 0.214·37-s + 0.871·38-s + 0.114·40-s + 1.43·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 2 | \( 1 - 61.1T + 2.04e3T^{2} \) |
| 5 | \( 1 + 3.36e3T + 4.88e7T^{2} \) |
| 7 | \( 1 - 1.07e4T + 1.97e9T^{2} \) |
| 11 | \( 1 + 2.39e4T + 2.85e11T^{2} \) |
| 13 | \( 1 - 2.37e6T + 1.79e12T^{2} \) |
| 17 | \( 1 + 8.22e6T + 3.42e13T^{2} \) |
| 19 | \( 1 - 6.96e6T + 1.16e14T^{2} \) |
| 23 | \( 1 + 3.92e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + 1.43e8T + 1.22e16T^{2} \) |
| 31 | \( 1 + 1.40e8T + 2.54e16T^{2} \) |
| 37 | \( 1 + 9.06e7T + 1.77e17T^{2} \) |
| 41 | \( 1 - 1.06e9T + 5.50e17T^{2} \) |
| 43 | \( 1 + 1.16e9T + 9.29e17T^{2} \) |
| 47 | \( 1 + 6.65e8T + 2.47e18T^{2} \) |
| 53 | \( 1 + 2.63e9T + 9.26e18T^{2} \) |
| 59 | \( 1 - 2.57e9T + 3.01e19T^{2} \) |
| 61 | \( 1 + 2.31e9T + 4.35e19T^{2} \) |
| 67 | \( 1 - 1.80e10T + 1.22e20T^{2} \) |
| 71 | \( 1 + 8.01e9T + 2.31e20T^{2} \) |
| 73 | \( 1 - 6.36e9T + 3.13e20T^{2} \) |
| 79 | \( 1 - 3.50e9T + 7.47e20T^{2} \) |
| 83 | \( 1 + 2.23e10T + 1.28e21T^{2} \) |
| 89 | \( 1 - 2.24e10T + 2.77e21T^{2} \) |
| 97 | \( 1 - 1.08e11T + 7.15e21T^{2} \) |
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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
−11.60983349352506936370438394926, −11.06539563221600365699292058208, −9.242836663997067945926406355839, −8.037076452147909184899761081368, −6.52454829766433596516462242240, −5.55432800964680019664717189091, −4.20152847494522520610452772497, −3.50689639278998180875459155448, −1.88294599321876723710916139221, 0,
1.88294599321876723710916139221, 3.50689639278998180875459155448, 4.20152847494522520610452772497, 5.55432800964680019664717189091, 6.52454829766433596516462242240, 8.037076452147909184899761081368, 9.242836663997067945926406355839, 11.06539563221600365699292058208, 11.60983349352506936370438394926
