| L(s) = 1 | + 0.618·2-s − 0.618·3-s − 1.61·4-s + 3.85·5-s − 0.381·6-s − 2.23·7-s − 2.23·8-s − 2.61·9-s + 2.38·10-s − 1.38·11-s + 1.00·12-s − 0.236·13-s − 1.38·14-s − 2.38·15-s + 1.85·16-s − 4.38·17-s − 1.61·18-s − 4.85·19-s − 6.23·20-s + 1.38·21-s − 0.854·22-s − 1.23·23-s + 1.38·24-s + 9.85·25-s − 0.145·26-s + 3.47·27-s + 3.61·28-s + ⋯ |
| L(s) = 1 | + 0.437·2-s − 0.356·3-s − 0.809·4-s + 1.72·5-s − 0.155·6-s − 0.845·7-s − 0.790·8-s − 0.872·9-s + 0.753·10-s − 0.416·11-s + 0.288·12-s − 0.0654·13-s − 0.369·14-s − 0.615·15-s + 0.463·16-s − 1.06·17-s − 0.381·18-s − 1.11·19-s − 1.39·20-s + 0.301·21-s − 0.182·22-s − 0.257·23-s + 0.282·24-s + 1.97·25-s − 0.0286·26-s + 0.668·27-s + 0.683·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 29 | \( 1 \) |
| good | 2 | \( 1 - 0.618T + 2T^{2} \) |
| 3 | \( 1 + 0.618T + 3T^{2} \) |
| 5 | \( 1 - 3.85T + 5T^{2} \) |
| 7 | \( 1 + 2.23T + 7T^{2} \) |
| 11 | \( 1 + 1.38T + 11T^{2} \) |
| 13 | \( 1 + 0.236T + 13T^{2} \) |
| 17 | \( 1 + 4.38T + 17T^{2} \) |
| 19 | \( 1 + 4.85T + 19T^{2} \) |
| 23 | \( 1 + 1.23T + 23T^{2} \) |
| 31 | \( 1 + 10.0T + 31T^{2} \) |
| 37 | \( 1 - 4.70T + 37T^{2} \) |
| 41 | \( 1 - 3.85T + 41T^{2} \) |
| 43 | \( 1 + 7.23T + 43T^{2} \) |
| 47 | \( 1 + 7T + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 - 6.09T + 59T^{2} \) |
| 61 | \( 1 + 0.618T + 61T^{2} \) |
| 67 | \( 1 + 1.52T + 67T^{2} \) |
| 71 | \( 1 - 10.4T + 71T^{2} \) |
| 73 | \( 1 + 13.7T + 73T^{2} \) |
| 79 | \( 1 + 6.09T + 79T^{2} \) |
| 83 | \( 1 + 9.94T + 83T^{2} \) |
| 89 | \( 1 - 4.70T + 89T^{2} \) |
| 97 | \( 1 - 3.56T + 97T^{2} \) |
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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
−9.711378688188144839945369600518, −9.097691096245418506206456907826, −8.405454958464993032384712060409, −6.72512851243885830412485772629, −6.03699646126926180927547171328, −5.50562936652875576665194517298, −4.56508001977196221481393428336, −3.18543640135487002394635092902, −2.13944074327352220016671955315, 0,
2.13944074327352220016671955315, 3.18543640135487002394635092902, 4.56508001977196221481393428336, 5.50562936652875576665194517298, 6.03699646126926180927547171328, 6.72512851243885830412485772629, 8.405454958464993032384712060409, 9.097691096245418506206456907826, 9.711378688188144839945369600518
