| L(s) = 1 | − 2.46·2-s + 4.05·4-s + 0.460·5-s − 0.593·7-s − 5.05·8-s − 1.13·10-s − 3.59·11-s + 0.593·13-s + 1.46·14-s + 4.32·16-s − 4.73·17-s − 19-s + 1.86·20-s + 8.84·22-s + 3.38·23-s − 4.78·25-s − 1.46·26-s − 2.40·28-s + 7.83·29-s − 1.59·31-s − 0.539·32-s + 11.6·34-s − 0.273·35-s − 4.86·37-s + 2.46·38-s − 2.32·40-s + 0.460·41-s + ⋯ |
| L(s) = 1 | − 1.73·2-s + 2.02·4-s + 0.205·5-s − 0.224·7-s − 1.78·8-s − 0.358·10-s − 1.08·11-s + 0.164·13-s + 0.390·14-s + 1.08·16-s − 1.14·17-s − 0.229·19-s + 0.417·20-s + 1.88·22-s + 0.705·23-s − 0.957·25-s − 0.286·26-s − 0.454·28-s + 1.45·29-s − 0.286·31-s − 0.0953·32-s + 1.99·34-s − 0.0462·35-s − 0.800·37-s + 0.399·38-s − 0.367·40-s + 0.0719·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 + 2.46T + 2T^{2} \) |
| 5 | \( 1 - 0.460T + 5T^{2} \) |
| 7 | \( 1 + 0.593T + 7T^{2} \) |
| 11 | \( 1 + 3.59T + 11T^{2} \) |
| 13 | \( 1 - 0.593T + 13T^{2} \) |
| 17 | \( 1 + 4.73T + 17T^{2} \) |
| 23 | \( 1 - 3.38T + 23T^{2} \) |
| 29 | \( 1 - 7.83T + 29T^{2} \) |
| 31 | \( 1 + 1.59T + 31T^{2} \) |
| 37 | \( 1 + 4.86T + 37T^{2} \) |
| 41 | \( 1 - 0.460T + 41T^{2} \) |
| 43 | \( 1 + 9.48T + 43T^{2} \) |
| 47 | \( 1 + 9T + 47T^{2} \) |
| 53 | \( 1 + 7.42T + 53T^{2} \) |
| 59 | \( 1 + 0.320T + 59T^{2} \) |
| 61 | \( 1 - 7.51T + 61T^{2} \) |
| 67 | \( 1 + 13.7T + 67T^{2} \) |
| 71 | \( 1 + 9.43T + 71T^{2} \) |
| 73 | \( 1 - 5.10T + 73T^{2} \) |
| 79 | \( 1 - 13.8T + 79T^{2} \) |
| 83 | \( 1 + 18.1T + 83T^{2} \) |
| 89 | \( 1 - 12.6T + 89T^{2} \) |
| 97 | \( 1 + 8.32T + 97T^{2} \) |
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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
−10.27843944574729558169425523336, −9.585448797860127252987186330144, −8.635118270220913268932176346297, −8.062379982221394521855513289605, −7.01395134050004250367768861727, −6.27005041201226705219627097684, −4.85519855316555856701450131345, −2.98041666186898465283490445185, −1.80779195613562411813798040004, 0,
1.80779195613562411813798040004, 2.98041666186898465283490445185, 4.85519855316555856701450131345, 6.27005041201226705219627097684, 7.01395134050004250367768861727, 8.062379982221394521855513289605, 8.635118270220913268932176346297, 9.585448797860127252987186330144, 10.27843944574729558169425523336
