| L(s) = 1 | − 0.976·2-s + 2.48·3-s − 1.04·4-s + 5-s − 2.42·6-s + 2.05·7-s + 2.97·8-s + 3.18·9-s − 0.976·10-s − 5.67·11-s − 2.60·12-s − 5.65·13-s − 2.00·14-s + 2.48·15-s − 0.809·16-s − 2.55·17-s − 3.11·18-s + 6.20·19-s − 1.04·20-s + 5.11·21-s + 5.54·22-s − 4.11·23-s + 7.40·24-s + 25-s + 5.51·26-s + 0.471·27-s − 2.15·28-s + ⋯ |
| L(s) = 1 | − 0.690·2-s + 1.43·3-s − 0.523·4-s + 0.447·5-s − 0.991·6-s + 0.777·7-s + 1.05·8-s + 1.06·9-s − 0.308·10-s − 1.71·11-s − 0.752·12-s − 1.56·13-s − 0.536·14-s + 0.642·15-s − 0.202·16-s − 0.619·17-s − 0.733·18-s + 1.42·19-s − 0.234·20-s + 1.11·21-s + 1.18·22-s − 0.858·23-s + 1.51·24-s + 0.200·25-s + 1.08·26-s + 0.0907·27-s − 0.406·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 29 | \( 1 \) |
| good | 2 | \( 1 + 0.976T + 2T^{2} \) |
| 3 | \( 1 - 2.48T + 3T^{2} \) |
| 7 | \( 1 - 2.05T + 7T^{2} \) |
| 11 | \( 1 + 5.67T + 11T^{2} \) |
| 13 | \( 1 + 5.65T + 13T^{2} \) |
| 17 | \( 1 + 2.55T + 17T^{2} \) |
| 19 | \( 1 - 6.20T + 19T^{2} \) |
| 23 | \( 1 + 4.11T + 23T^{2} \) |
| 31 | \( 1 + 0.0152T + 31T^{2} \) |
| 37 | \( 1 - 1.55T + 37T^{2} \) |
| 41 | \( 1 - 2.39T + 41T^{2} \) |
| 43 | \( 1 + 8.83T + 43T^{2} \) |
| 47 | \( 1 - 2.04T + 47T^{2} \) |
| 53 | \( 1 - 0.225T + 53T^{2} \) |
| 59 | \( 1 - 11.6T + 59T^{2} \) |
| 61 | \( 1 + 8.92T + 61T^{2} \) |
| 67 | \( 1 + 14.8T + 67T^{2} \) |
| 71 | \( 1 + 0.755T + 71T^{2} \) |
| 73 | \( 1 + 1.50T + 73T^{2} \) |
| 79 | \( 1 + 8.03T + 79T^{2} \) |
| 83 | \( 1 + 3.54T + 83T^{2} \) |
| 89 | \( 1 + 4.85T + 89T^{2} \) |
| 97 | \( 1 + 3.67T + 97T^{2} \) |
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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
−8.075804342423567208160670955584, −7.64746998121193715051740505789, −7.17340216489551690181667690359, −5.56269083238057003772290452989, −4.97421893178366825443400350608, −4.30426576554823597798671299951, −3.05716811826567700731819249045, −2.40028596084873176662339744613, −1.61266858504383034718000330026, 0,
1.61266858504383034718000330026, 2.40028596084873176662339744613, 3.05716811826567700731819249045, 4.30426576554823597798671299951, 4.97421893178366825443400350608, 5.56269083238057003772290452989, 7.17340216489551690181667690359, 7.64746998121193715051740505789, 8.075804342423567208160670955584
