| L(s) = 1 | + 6.27e8·2-s − 1.91e18·4-s − 2.33e20·5-s + 6.25e25·7-s − 2.64e27·8-s − 1.46e29·10-s + 9.05e31·11-s + 7.79e33·13-s + 3.92e34·14-s + 2.74e36·16-s + 1.64e37·17-s + 1.45e39·19-s + 4.47e38·20-s + 5.68e40·22-s + 1.28e41·23-s − 4.28e42·25-s + 4.89e42·26-s − 1.19e44·28-s − 1.23e44·29-s + 2.17e45·31-s + 7.82e45·32-s + 1.03e46·34-s − 1.46e46·35-s + 5.13e47·37-s + 9.12e47·38-s + 6.19e47·40-s − 1.44e49·41-s + ⋯ |
| L(s) = 1 | + 0.413·2-s − 0.829·4-s − 0.112·5-s + 1.04·7-s − 0.756·8-s − 0.0464·10-s + 1.56·11-s + 0.824·13-s + 0.433·14-s + 0.516·16-s + 0.485·17-s + 1.44·19-s + 0.0931·20-s + 0.646·22-s + 0.377·23-s − 0.987·25-s + 0.341·26-s − 0.869·28-s − 0.309·29-s + 0.710·31-s + 0.969·32-s + 0.200·34-s − 0.117·35-s + 0.758·37-s + 0.597·38-s + 0.0849·40-s − 0.930·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(62-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+61/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(31)\) |
\(\approx\) |
\(3.419040613\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.419040613\) |
| \(L(\frac{63}{2})\) |
|
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 | \( 1 - 6.27e8T + 2.30e18T^{2} \) |
| 5 | \( 1 + 2.33e20T + 4.33e42T^{2} \) |
| 7 | \( 1 - 6.25e25T + 3.55e51T^{2} \) |
| 11 | \( 1 - 9.05e31T + 3.34e63T^{2} \) |
| 13 | \( 1 - 7.79e33T + 8.92e67T^{2} \) |
| 17 | \( 1 - 1.64e37T + 1.14e75T^{2} \) |
| 19 | \( 1 - 1.45e39T + 1.00e78T^{2} \) |
| 23 | \( 1 - 1.28e41T + 1.16e83T^{2} \) |
| 29 | \( 1 + 1.23e44T + 1.60e89T^{2} \) |
| 31 | \( 1 - 2.17e45T + 9.39e90T^{2} \) |
| 37 | \( 1 - 5.13e47T + 4.57e95T^{2} \) |
| 41 | \( 1 + 1.44e49T + 2.39e98T^{2} \) |
| 43 | \( 1 + 4.68e49T + 4.38e99T^{2} \) |
| 47 | \( 1 - 1.55e51T + 9.95e101T^{2} \) |
| 53 | \( 1 + 2.61e52T + 1.51e105T^{2} \) |
| 59 | \( 1 + 5.85e53T + 1.05e108T^{2} \) |
| 61 | \( 1 + 6.99e53T + 8.03e108T^{2} \) |
| 67 | \( 1 + 5.65e55T + 2.45e111T^{2} \) |
| 71 | \( 1 - 4.76e56T + 8.44e112T^{2} \) |
| 73 | \( 1 - 4.87e56T + 4.59e113T^{2} \) |
| 79 | \( 1 + 6.21e57T + 5.69e115T^{2} \) |
| 83 | \( 1 - 1.18e58T + 1.15e117T^{2} \) |
| 89 | \( 1 - 8.44e58T + 8.18e118T^{2} \) |
| 97 | \( 1 - 8.57e59T + 1.55e121T^{2} \) |
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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
−11.37880179766527268538884458266, −9.716169797769431589775003507717, −8.774305737131621537646990866820, −7.72173273536138967974532030840, −6.18332763978163277746744847638, −5.12769074796375870113386579393, −4.12013928789620171102887829240, −3.31452808991492456303728547676, −1.52223712205931147902167428349, −0.817504002949265572488123962499,
0.817504002949265572488123962499, 1.52223712205931147902167428349, 3.31452808991492456303728547676, 4.12013928789620171102887829240, 5.12769074796375870113386579393, 6.18332763978163277746744847638, 7.72173273536138967974532030840, 8.774305737131621537646990866820, 9.716169797769431589775003507717, 11.37880179766527268538884458266
