| L(s) = 1 | − 0.570·2-s + 0.725·3-s − 1.67·4-s − 5-s − 0.414·6-s − 7-s + 2.09·8-s − 2.47·9-s + 0.570·10-s − 4.87·11-s − 1.21·12-s − 2.95·13-s + 0.570·14-s − 0.725·15-s + 2.15·16-s + 1.20·17-s + 1.41·18-s + 8.00·19-s + 1.67·20-s − 0.725·21-s + 2.78·22-s − 1.22·23-s + 1.52·24-s + 25-s + 1.68·26-s − 3.97·27-s + 1.67·28-s + ⋯ |
| L(s) = 1 | − 0.403·2-s + 0.418·3-s − 0.837·4-s − 0.447·5-s − 0.169·6-s − 0.377·7-s + 0.741·8-s − 0.824·9-s + 0.180·10-s − 1.46·11-s − 0.350·12-s − 0.820·13-s + 0.152·14-s − 0.187·15-s + 0.537·16-s + 0.293·17-s + 0.332·18-s + 1.83·19-s + 0.374·20-s − 0.158·21-s + 0.593·22-s − 0.255·23-s + 0.310·24-s + 0.200·25-s + 0.331·26-s − 0.764·27-s + 0.316·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 229 | \( 1 + T \) |
| good | 2 | \( 1 + 0.570T + 2T^{2} \) |
| 3 | \( 1 - 0.725T + 3T^{2} \) |
| 11 | \( 1 + 4.87T + 11T^{2} \) |
| 13 | \( 1 + 2.95T + 13T^{2} \) |
| 17 | \( 1 - 1.20T + 17T^{2} \) |
| 19 | \( 1 - 8.00T + 19T^{2} \) |
| 23 | \( 1 + 1.22T + 23T^{2} \) |
| 29 | \( 1 - 8.16T + 29T^{2} \) |
| 31 | \( 1 + 1.32T + 31T^{2} \) |
| 37 | \( 1 + 1.47T + 37T^{2} \) |
| 41 | \( 1 - 4.34T + 41T^{2} \) |
| 43 | \( 1 - 2.89T + 43T^{2} \) |
| 47 | \( 1 - 3.81T + 47T^{2} \) |
| 53 | \( 1 - 3.83T + 53T^{2} \) |
| 59 | \( 1 + 14.5T + 59T^{2} \) |
| 61 | \( 1 - 5.02T + 61T^{2} \) |
| 67 | \( 1 - 9.73T + 67T^{2} \) |
| 71 | \( 1 + 4.33T + 71T^{2} \) |
| 73 | \( 1 + 4.95T + 73T^{2} \) |
| 79 | \( 1 - 11.8T + 79T^{2} \) |
| 83 | \( 1 - 2.98T + 83T^{2} \) |
| 89 | \( 1 - 15.5T + 89T^{2} \) |
| 97 | \( 1 + 17.7T + 97T^{2} \) |
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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
−7.77711165315882331610391269190, −7.16725095060149214577592128222, −5.95634317165824510412848270348, −5.23429053215147256197135377055, −4.83710246152921167583122153433, −3.77238165127960587621517097889, −3.03674095566301872314313469169, −2.45826695758900635982548671018, −0.926440690902669775677734112095, 0,
0.926440690902669775677734112095, 2.45826695758900635982548671018, 3.03674095566301872314313469169, 3.77238165127960587621517097889, 4.83710246152921167583122153433, 5.23429053215147256197135377055, 5.95634317165824510412848270348, 7.16725095060149214577592128222, 7.77711165315882331610391269190
