| L(s) = 1 | − 0.938·2-s + 2.04·3-s − 1.11·4-s + 3.68·5-s − 1.92·6-s − 2.71·7-s + 2.92·8-s + 1.18·9-s − 3.46·10-s − 2.41·11-s − 2.28·12-s + 5.06·13-s + 2.55·14-s + 7.54·15-s − 0.509·16-s + 2.79·17-s − 1.11·18-s + 4.67·19-s − 4.12·20-s − 5.55·21-s + 2.26·22-s + 2.15·23-s + 5.98·24-s + 8.60·25-s − 4.75·26-s − 3.71·27-s + 3.04·28-s + ⋯ |
| L(s) = 1 | − 0.663·2-s + 1.18·3-s − 0.559·4-s + 1.64·5-s − 0.783·6-s − 1.02·7-s + 1.03·8-s + 0.395·9-s − 1.09·10-s − 0.728·11-s − 0.660·12-s + 1.40·13-s + 0.681·14-s + 1.94·15-s − 0.127·16-s + 0.678·17-s − 0.262·18-s + 1.07·19-s − 0.922·20-s − 1.21·21-s + 0.483·22-s + 0.449·23-s + 1.22·24-s + 1.72·25-s − 0.932·26-s − 0.714·27-s + 0.574·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 619 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 619 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.656202764\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.656202764\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 619 | \( 1 - T \) |
| good | 2 | \( 1 + 0.938T + 2T^{2} \) |
| 3 | \( 1 - 2.04T + 3T^{2} \) |
| 5 | \( 1 - 3.68T + 5T^{2} \) |
| 7 | \( 1 + 2.71T + 7T^{2} \) |
| 11 | \( 1 + 2.41T + 11T^{2} \) |
| 13 | \( 1 - 5.06T + 13T^{2} \) |
| 17 | \( 1 - 2.79T + 17T^{2} \) |
| 19 | \( 1 - 4.67T + 19T^{2} \) |
| 23 | \( 1 - 2.15T + 23T^{2} \) |
| 29 | \( 1 - 0.405T + 29T^{2} \) |
| 31 | \( 1 + 2.87T + 31T^{2} \) |
| 37 | \( 1 - 4.37T + 37T^{2} \) |
| 41 | \( 1 + 2.91T + 41T^{2} \) |
| 43 | \( 1 - 4.39T + 43T^{2} \) |
| 47 | \( 1 + 4.68T + 47T^{2} \) |
| 53 | \( 1 - 5.90T + 53T^{2} \) |
| 59 | \( 1 - 2.92T + 59T^{2} \) |
| 61 | \( 1 + 8.81T + 61T^{2} \) |
| 67 | \( 1 - 6.63T + 67T^{2} \) |
| 71 | \( 1 - 1.75T + 71T^{2} \) |
| 73 | \( 1 - 11.6T + 73T^{2} \) |
| 79 | \( 1 + 17.0T + 79T^{2} \) |
| 83 | \( 1 + 9.17T + 83T^{2} \) |
| 89 | \( 1 + 10.6T + 89T^{2} \) |
| 97 | \( 1 + 6.48T + 97T^{2} \) |
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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
−10.01704971179075392654616799481, −9.698605910601622948978197970639, −9.000945006061346155400715381398, −8.351056046801468537396618282440, −7.32884282358836983287553381388, −6.06780888894087029413180256663, −5.29375584344496105348722858488, −3.62389188238814567424070208609, −2.72072246623048306659908042891, −1.35060995591817130036840513814,
1.35060995591817130036840513814, 2.72072246623048306659908042891, 3.62389188238814567424070208609, 5.29375584344496105348722858488, 6.06780888894087029413180256663, 7.32884282358836983287553381388, 8.351056046801468537396618282440, 9.000945006061346155400715381398, 9.698605910601622948978197970639, 10.01704971179075392654616799481
