| L(s) = 1 | + 50.9·2-s + 548.·4-s − 7.10e3·5-s − 7.89e4·7-s − 7.64e4·8-s − 3.61e5·10-s + 4.12e5·11-s − 2.71e5·13-s − 4.02e6·14-s − 5.01e6·16-s − 1.90e6·17-s + 1.56e7·19-s − 3.89e6·20-s + 2.10e7·22-s + 4.27e7·23-s + 1.61e6·25-s − 1.38e7·26-s − 4.32e7·28-s + 5.17e7·29-s + 1.31e8·31-s − 9.91e7·32-s − 9.69e7·34-s + 5.60e8·35-s − 4.81e8·37-s + 7.96e8·38-s + 5.42e8·40-s + 9.42e8·41-s + ⋯ |
| L(s) = 1 | + 1.12·2-s + 0.267·4-s − 1.01·5-s − 1.77·7-s − 0.824·8-s − 1.14·10-s + 0.772·11-s − 0.202·13-s − 1.99·14-s − 1.19·16-s − 0.325·17-s + 1.44·19-s − 0.272·20-s + 0.869·22-s + 1.38·23-s + 0.0330·25-s − 0.228·26-s − 0.475·28-s + 0.468·29-s + 0.823·31-s − 0.522·32-s − 0.366·34-s + 1.80·35-s − 1.14·37-s + 1.63·38-s + 0.838·40-s + 1.27·41-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
| \(L(6)\) |
\(\approx\) |
\(1.677192381\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.677192381\) |
| \(L(\frac{13}{2})\) |
|
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 | \( 1 - 50.9T + 2.04e3T^{2} \) |
| 5 | \( 1 + 7.10e3T + 4.88e7T^{2} \) |
| 7 | \( 1 + 7.89e4T + 1.97e9T^{2} \) |
| 11 | \( 1 - 4.12e5T + 2.85e11T^{2} \) |
| 13 | \( 1 + 2.71e5T + 1.79e12T^{2} \) |
| 17 | \( 1 + 1.90e6T + 3.42e13T^{2} \) |
| 19 | \( 1 - 1.56e7T + 1.16e14T^{2} \) |
| 23 | \( 1 - 4.27e7T + 9.52e14T^{2} \) |
| 29 | \( 1 - 5.17e7T + 1.22e16T^{2} \) |
| 31 | \( 1 - 1.31e8T + 2.54e16T^{2} \) |
| 37 | \( 1 + 4.81e8T + 1.77e17T^{2} \) |
| 41 | \( 1 - 9.42e8T + 5.50e17T^{2} \) |
| 43 | \( 1 + 1.59e9T + 9.29e17T^{2} \) |
| 47 | \( 1 + 2.19e9T + 2.47e18T^{2} \) |
| 53 | \( 1 - 3.39e9T + 9.26e18T^{2} \) |
| 59 | \( 1 + 6.45e9T + 3.01e19T^{2} \) |
| 61 | \( 1 + 1.80e9T + 4.35e19T^{2} \) |
| 67 | \( 1 + 6.18e9T + 1.22e20T^{2} \) |
| 71 | \( 1 - 8.00e9T + 2.31e20T^{2} \) |
| 73 | \( 1 - 8.88e9T + 3.13e20T^{2} \) |
| 79 | \( 1 - 4.10e10T + 7.47e20T^{2} \) |
| 83 | \( 1 + 5.33e10T + 1.28e21T^{2} \) |
| 89 | \( 1 + 3.21e10T + 2.77e21T^{2} \) |
| 97 | \( 1 - 1.27e11T + 7.15e21T^{2} \) |
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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
−12.25599133537188966467273369404, −11.54241610495673363440653023194, −9.822261145094996110315952572470, −8.878089146800479149132870943601, −7.13906830472322589286191903814, −6.25166460342265692349136504551, −4.83504872227111191040380877454, −3.58738829713565576154511474915, −3.06581066072467803715948490135, −0.56761593365112408435218428316,
0.56761593365112408435218428316, 3.06581066072467803715948490135, 3.58738829713565576154511474915, 4.83504872227111191040380877454, 6.25166460342265692349136504551, 7.13906830472322589286191903814, 8.878089146800479149132870943601, 9.822261145094996110315952572470, 11.54241610495673363440653023194, 12.25599133537188966467273369404
