| L(s) = 1 | + 3.48e3·2-s + 5.17e5·3-s + 3.74e6·4-s − 4.88e7·5-s + 1.80e9·6-s + 8.64e9·7-s − 1.61e10·8-s + 1.73e11·9-s − 1.70e11·10-s + 4.59e11·11-s + 1.93e12·12-s − 2.79e12·13-s + 3.01e13·14-s − 2.52e13·15-s − 8.77e13·16-s − 1.40e14·17-s + 6.05e14·18-s + 7.82e14·19-s − 1.82e14·20-s + 4.47e15·21-s + 1.60e15·22-s − 3.06e15·23-s − 8.37e15·24-s + 2.38e15·25-s − 9.72e15·26-s + 4.11e16·27-s + 3.23e16·28-s + ⋯ |
| L(s) = 1 | + 1.20·2-s + 1.68·3-s + 0.446·4-s − 0.447·5-s + 2.02·6-s + 1.65·7-s − 0.666·8-s + 1.84·9-s − 0.537·10-s + 0.485·11-s + 0.752·12-s − 0.432·13-s + 1.98·14-s − 0.754·15-s − 1.24·16-s − 0.994·17-s + 2.21·18-s + 1.54·19-s − 0.199·20-s + 2.78·21-s + 0.583·22-s − 0.671·23-s − 1.12·24-s + 0.200·25-s − 0.519·26-s + 1.42·27-s + 0.737·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(24-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+23/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(12)\) |
\(\approx\) |
\(5.665472109\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.665472109\) |
| \(L(\frac{25}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 4.88e7T \) |
| good | 2 | \( 1 - 3.48e3T + 8.38e6T^{2} \) |
| 3 | \( 1 - 5.17e5T + 9.41e10T^{2} \) |
| 7 | \( 1 - 8.64e9T + 2.73e19T^{2} \) |
| 11 | \( 1 - 4.59e11T + 8.95e23T^{2} \) |
| 13 | \( 1 + 2.79e12T + 4.17e25T^{2} \) |
| 17 | \( 1 + 1.40e14T + 1.99e28T^{2} \) |
| 19 | \( 1 - 7.82e14T + 2.57e29T^{2} \) |
| 23 | \( 1 + 3.06e15T + 2.08e31T^{2} \) |
| 29 | \( 1 + 9.69e16T + 4.31e33T^{2} \) |
| 31 | \( 1 - 1.54e16T + 2.00e34T^{2} \) |
| 37 | \( 1 + 4.47e17T + 1.17e36T^{2} \) |
| 41 | \( 1 + 4.27e18T + 1.24e37T^{2} \) |
| 43 | \( 1 - 3.50e18T + 3.71e37T^{2} \) |
| 47 | \( 1 + 1.79e19T + 2.87e38T^{2} \) |
| 53 | \( 1 - 4.84e19T + 4.55e39T^{2} \) |
| 59 | \( 1 + 1.49e20T + 5.36e40T^{2} \) |
| 61 | \( 1 + 1.86e20T + 1.15e41T^{2} \) |
| 67 | \( 1 + 1.62e21T + 9.99e41T^{2} \) |
| 71 | \( 1 - 1.03e21T + 3.79e42T^{2} \) |
| 73 | \( 1 - 4.50e20T + 7.18e42T^{2} \) |
| 79 | \( 1 - 7.76e21T + 4.42e43T^{2} \) |
| 83 | \( 1 - 6.10e21T + 1.37e44T^{2} \) |
| 89 | \( 1 - 4.80e22T + 6.85e44T^{2} \) |
| 97 | \( 1 - 1.13e23T + 4.96e45T^{2} \) |
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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
−18.18136364870800346423983539182, −15.30006025531718162737342401348, −14.47617905979983717007872202670, −13.58416652989357594196091461228, −11.74481850817345577005622209115, −9.004829544036626403712619876689, −7.62829913880880421689948450465, −4.77080032599463609750993073380, −3.55176220364813468585921704467, −1.95993558721170510174804550017,
1.95993558721170510174804550017, 3.55176220364813468585921704467, 4.77080032599463609750993073380, 7.62829913880880421689948450465, 9.004829544036626403712619876689, 11.74481850817345577005622209115, 13.58416652989357594196091461228, 14.47617905979983717007872202670, 15.30006025531718162737342401348, 18.18136364870800346423983539182
