| L(s) = 1 | − 1.24·2-s − 0.452·4-s + 2.65·5-s − 1.13·7-s + 3.05·8-s − 3.30·10-s − 0.387·11-s − 3.92·13-s + 1.41·14-s − 2.89·16-s − 5.40·17-s + 0.820·19-s − 1.20·20-s + 0.482·22-s − 8.63·23-s + 2.07·25-s + 4.88·26-s + 0.514·28-s + 8.20·29-s + 7.24·31-s − 2.50·32-s + 6.71·34-s − 3.02·35-s − 7.63·37-s − 1.02·38-s + 8.11·40-s − 1.27·41-s + ⋯ |
| L(s) = 1 | − 0.879·2-s − 0.226·4-s + 1.18·5-s − 0.429·7-s + 1.07·8-s − 1.04·10-s − 0.116·11-s − 1.08·13-s + 0.378·14-s − 0.722·16-s − 1.31·17-s + 0.188·19-s − 0.269·20-s + 0.102·22-s − 1.80·23-s + 0.414·25-s + 0.958·26-s + 0.0972·28-s + 1.52·29-s + 1.30·31-s − 0.442·32-s + 1.15·34-s − 0.511·35-s − 1.25·37-s − 0.165·38-s + 1.28·40-s − 0.198·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1143 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 127 | \( 1 - T \) |
| good | 2 | \( 1 + 1.24T + 2T^{2} \) |
| 5 | \( 1 - 2.65T + 5T^{2} \) |
| 7 | \( 1 + 1.13T + 7T^{2} \) |
| 11 | \( 1 + 0.387T + 11T^{2} \) |
| 13 | \( 1 + 3.92T + 13T^{2} \) |
| 17 | \( 1 + 5.40T + 17T^{2} \) |
| 19 | \( 1 - 0.820T + 19T^{2} \) |
| 23 | \( 1 + 8.63T + 23T^{2} \) |
| 29 | \( 1 - 8.20T + 29T^{2} \) |
| 31 | \( 1 - 7.24T + 31T^{2} \) |
| 37 | \( 1 + 7.63T + 37T^{2} \) |
| 41 | \( 1 + 1.27T + 41T^{2} \) |
| 43 | \( 1 + 4.31T + 43T^{2} \) |
| 47 | \( 1 - 7.17T + 47T^{2} \) |
| 53 | \( 1 + 12.0T + 53T^{2} \) |
| 59 | \( 1 - 7.80T + 59T^{2} \) |
| 61 | \( 1 - 3.19T + 61T^{2} \) |
| 67 | \( 1 + 6.56T + 67T^{2} \) |
| 71 | \( 1 + 14.2T + 71T^{2} \) |
| 73 | \( 1 + 0.473T + 73T^{2} \) |
| 79 | \( 1 + 5.31T + 79T^{2} \) |
| 83 | \( 1 - 5.39T + 83T^{2} \) |
| 89 | \( 1 + 10.1T + 89T^{2} \) |
| 97 | \( 1 - 6.53T + 97T^{2} \) |
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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
−9.520934096978615862774121334510, −8.715943276938461397787039719183, −7.991712328010445553178877362663, −6.90823817364104458472864948274, −6.20030591039785849908097387926, −5.08715355577981080538833778650, −4.29253281897153605862682904789, −2.66174228035696327619455563202, −1.70743269511736537937152619436, 0,
1.70743269511736537937152619436, 2.66174228035696327619455563202, 4.29253281897153605862682904789, 5.08715355577981080538833778650, 6.20030591039785849908097387926, 6.90823817364104458472864948274, 7.991712328010445553178877362663, 8.715943276938461397787039719183, 9.520934096978615862774121334510
