| L(s) = 1 | − 0.676·2-s − 1.54·4-s + 1.80·5-s − 4.44·7-s + 2.39·8-s − 1.22·10-s + 1.20·11-s − 3.05·13-s + 3.00·14-s + 1.46·16-s + 6.57·17-s − 2.24·19-s − 2.78·20-s − 0.814·22-s + 23-s − 1.73·25-s + 2.06·26-s + 6.85·28-s − 29-s + 2.34·31-s − 5.78·32-s − 4.44·34-s − 8.02·35-s − 8.63·37-s + 1.51·38-s + 4.32·40-s + 1.57·41-s + ⋯ |
| L(s) = 1 | − 0.478·2-s − 0.771·4-s + 0.807·5-s − 1.68·7-s + 0.847·8-s − 0.386·10-s + 0.362·11-s − 0.847·13-s + 0.803·14-s + 0.365·16-s + 1.59·17-s − 0.514·19-s − 0.622·20-s − 0.173·22-s + 0.208·23-s − 0.347·25-s + 0.405·26-s + 1.29·28-s − 0.185·29-s + 0.422·31-s − 1.02·32-s − 0.762·34-s − 1.35·35-s − 1.41·37-s + 0.245·38-s + 0.684·40-s + 0.245·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + 0.676T + 2T^{2} \) |
| 5 | \( 1 - 1.80T + 5T^{2} \) |
| 7 | \( 1 + 4.44T + 7T^{2} \) |
| 11 | \( 1 - 1.20T + 11T^{2} \) |
| 13 | \( 1 + 3.05T + 13T^{2} \) |
| 17 | \( 1 - 6.57T + 17T^{2} \) |
| 19 | \( 1 + 2.24T + 19T^{2} \) |
| 31 | \( 1 - 2.34T + 31T^{2} \) |
| 37 | \( 1 + 8.63T + 37T^{2} \) |
| 41 | \( 1 - 1.57T + 41T^{2} \) |
| 43 | \( 1 - 6.47T + 43T^{2} \) |
| 47 | \( 1 - 8.54T + 47T^{2} \) |
| 53 | \( 1 + 11.5T + 53T^{2} \) |
| 59 | \( 1 - 6.15T + 59T^{2} \) |
| 61 | \( 1 + 0.352T + 61T^{2} \) |
| 67 | \( 1 - 1.02T + 67T^{2} \) |
| 71 | \( 1 - 3.08T + 71T^{2} \) |
| 73 | \( 1 + 8.06T + 73T^{2} \) |
| 79 | \( 1 - 4.13T + 79T^{2} \) |
| 83 | \( 1 - 1.05T + 83T^{2} \) |
| 89 | \( 1 + 2.79T + 89T^{2} \) |
| 97 | \( 1 - 8.49T + 97T^{2} \) |
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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
−7.68743809936363390624667958418, −7.12241785513425005415183788246, −6.22843574804512263688183348248, −5.69937513104064743628864123156, −4.93622212909709943461985326306, −3.90819335715407219962239387319, −3.27526444298335047285293310298, −2.29173019350000604824455197770, −1.09648818487166048878344542753, 0,
1.09648818487166048878344542753, 2.29173019350000604824455197770, 3.27526444298335047285293310298, 3.90819335715407219962239387319, 4.93622212909709943461985326306, 5.69937513104064743628864123156, 6.22843574804512263688183348248, 7.12241785513425005415183788246, 7.68743809936363390624667958418
