| L(s) = 1 | − 4·2-s + 8·3-s + 16·4-s − 25·5-s − 32·6-s + 199·7-s − 64·8-s − 179·9-s + 100·10-s + 150·11-s + 128·12-s − 1.20e3·13-s − 796·14-s − 200·15-s + 256·16-s + 735·17-s + 716·18-s − 22·19-s − 400·20-s + 1.59e3·21-s − 600·22-s − 529·23-s − 512·24-s + 625·25-s + 4.80e3·26-s − 3.37e3·27-s + 3.18e3·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.513·3-s + 1/2·4-s − 0.447·5-s − 0.362·6-s + 1.53·7-s − 0.353·8-s − 0.736·9-s + 0.316·10-s + 0.373·11-s + 0.256·12-s − 1.97·13-s − 1.08·14-s − 0.229·15-s + 1/4·16-s + 0.616·17-s + 0.520·18-s − 0.0139·19-s − 0.223·20-s + 0.787·21-s − 0.264·22-s − 0.208·23-s − 0.181·24-s + 1/5·25-s + 1.39·26-s − 0.891·27-s + 0.767·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{2} T \) |
| 5 | \( 1 + p^{2} T \) |
| 23 | \( 1 + p^{2} T \) |
| good | 3 | \( 1 - 8 T + p^{5} T^{2} \) |
| 7 | \( 1 - 199 T + p^{5} T^{2} \) |
| 11 | \( 1 - 150 T + p^{5} T^{2} \) |
| 13 | \( 1 + 1202 T + p^{5} T^{2} \) |
| 17 | \( 1 - 735 T + p^{5} T^{2} \) |
| 19 | \( 1 + 22 T + p^{5} T^{2} \) |
| 29 | \( 1 + 5525 T + p^{5} T^{2} \) |
| 31 | \( 1 + 95 T + p^{5} T^{2} \) |
| 37 | \( 1 + 397 T + p^{5} T^{2} \) |
| 41 | \( 1 - 20633 T + p^{5} T^{2} \) |
| 43 | \( 1 + 11384 T + p^{5} T^{2} \) |
| 47 | \( 1 - 1992 T + p^{5} T^{2} \) |
| 53 | \( 1 + 7349 T + p^{5} T^{2} \) |
| 59 | \( 1 + 23827 T + p^{5} T^{2} \) |
| 61 | \( 1 + 44016 T + p^{5} T^{2} \) |
| 67 | \( 1 + 37713 T + p^{5} T^{2} \) |
| 71 | \( 1 + 50057 T + p^{5} T^{2} \) |
| 73 | \( 1 + 16698 T + p^{5} T^{2} \) |
| 79 | \( 1 + 31004 T + p^{5} T^{2} \) |
| 83 | \( 1 + 70077 T + p^{5} T^{2} \) |
| 89 | \( 1 - 7676 T + p^{5} T^{2} \) |
| 97 | \( 1 + 150094 T + p^{5} T^{2} \) |
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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
−10.92478931862965906928124621387, −9.695287932727095481350942959946, −8.815280041525245543339260129928, −7.79645185320190228978750908186, −7.43650054493850525090907900767, −5.62837358611476366990358242167, −4.46315318006215781964724633901, −2.83223059447641587728913485020, −1.67496418093733385427697234596, 0,
1.67496418093733385427697234596, 2.83223059447641587728913485020, 4.46315318006215781964724633901, 5.62837358611476366990358242167, 7.43650054493850525090907900767, 7.79645185320190228978750908186, 8.815280041525245543339260129928, 9.695287932727095481350942959946, 10.92478931862965906928124621387
