| L(s) = 1 | − 1.59e3·2-s + 9.39e3·3-s + 4.39e5·4-s − 1.49e7·6-s + 1.90e8·7-s + 2.64e9·8-s − 1.03e10·9-s + 5.63e9·11-s + 4.13e9·12-s − 9.01e10·13-s − 3.03e11·14-s − 5.12e12·16-s + 9.42e12·17-s + 1.65e13·18-s − 2.01e13·19-s + 1.78e12·21-s − 8.97e12·22-s − 1.81e14·23-s + 2.48e13·24-s + 1.43e14·26-s − 1.95e14·27-s + 8.36e13·28-s + 1.06e15·29-s + 5.13e15·31-s + 2.62e15·32-s + 5.29e13·33-s − 1.50e16·34-s + ⋯ |
| L(s) = 1 | − 1.09·2-s + 0.0918·3-s + 0.209·4-s − 0.101·6-s + 0.254·7-s + 0.869·8-s − 0.991·9-s + 0.0655·11-s + 0.0192·12-s − 0.181·13-s − 0.280·14-s − 1.16·16-s + 1.13·17-s + 1.09·18-s − 0.754·19-s + 0.0234·21-s − 0.0720·22-s − 0.914·23-s + 0.0798·24-s + 0.199·26-s − 0.183·27-s + 0.0533·28-s + 0.469·29-s + 1.12·31-s + 0.412·32-s + 0.00601·33-s − 1.24·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(11)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{23}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| good | 2 | \( 1 + 1.59e3T + 2.09e6T^{2} \) |
| 3 | \( 1 - 9.39e3T + 1.04e10T^{2} \) |
| 7 | \( 1 - 1.90e8T + 5.58e17T^{2} \) |
| 11 | \( 1 - 5.63e9T + 7.40e21T^{2} \) |
| 13 | \( 1 + 9.01e10T + 2.47e23T^{2} \) |
| 17 | \( 1 - 9.42e12T + 6.90e25T^{2} \) |
| 19 | \( 1 + 2.01e13T + 7.14e26T^{2} \) |
| 23 | \( 1 + 1.81e14T + 3.94e28T^{2} \) |
| 29 | \( 1 - 1.06e15T + 5.13e30T^{2} \) |
| 31 | \( 1 - 5.13e15T + 2.08e31T^{2} \) |
| 37 | \( 1 - 3.84e16T + 8.55e32T^{2} \) |
| 41 | \( 1 - 1.47e17T + 7.38e33T^{2} \) |
| 43 | \( 1 + 8.58e16T + 2.00e34T^{2} \) |
| 47 | \( 1 - 4.33e17T + 1.30e35T^{2} \) |
| 53 | \( 1 - 1.54e18T + 1.62e36T^{2} \) |
| 59 | \( 1 + 3.53e18T + 1.54e37T^{2} \) |
| 61 | \( 1 + 5.04e18T + 3.10e37T^{2} \) |
| 67 | \( 1 + 2.20e19T + 2.22e38T^{2} \) |
| 71 | \( 1 + 4.58e19T + 7.52e38T^{2} \) |
| 73 | \( 1 - 2.75e19T + 1.34e39T^{2} \) |
| 79 | \( 1 + 5.50e19T + 7.08e39T^{2} \) |
| 83 | \( 1 - 9.94e19T + 1.99e40T^{2} \) |
| 89 | \( 1 + 5.03e20T + 8.65e40T^{2} \) |
| 97 | \( 1 + 7.08e20T + 5.27e41T^{2} \) |
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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
−12.05959129401259765308302070039, −10.77203293836046643085214173563, −9.666167917173116421097110092018, −8.501249305692260721884639500921, −7.68079020831484390059327042329, −5.95168879262124352479868515311, −4.37521148056138028426325615811, −2.61645334314265723948834828754, −1.16562518689319448290643601396, 0,
1.16562518689319448290643601396, 2.61645334314265723948834828754, 4.37521148056138028426325615811, 5.95168879262124352479868515311, 7.68079020831484390059327042329, 8.501249305692260721884639500921, 9.666167917173116421097110092018, 10.77203293836046643085214173563, 12.05959129401259765308302070039
