| L(s) = 1 | + 2.46·2-s − 0.157·3-s + 4.08·4-s + 2.37·5-s − 0.389·6-s + 4.70·7-s + 5.13·8-s − 2.97·9-s + 5.86·10-s − 3.89·11-s − 0.644·12-s − 6.62·13-s + 11.6·14-s − 0.375·15-s + 4.49·16-s − 5.95·17-s − 7.33·18-s − 0.395·19-s + 9.70·20-s − 0.743·21-s − 9.60·22-s − 1.84·23-s − 0.810·24-s + 0.653·25-s − 16.3·26-s + 0.943·27-s + 19.2·28-s + ⋯ |
| L(s) = 1 | + 1.74·2-s − 0.0911·3-s + 2.04·4-s + 1.06·5-s − 0.158·6-s + 1.77·7-s + 1.81·8-s − 0.991·9-s + 1.85·10-s − 1.17·11-s − 0.185·12-s − 1.83·13-s + 3.10·14-s − 0.0969·15-s + 1.12·16-s − 1.44·17-s − 1.72·18-s − 0.0906·19-s + 2.16·20-s − 0.162·21-s − 2.04·22-s − 0.385·23-s − 0.165·24-s + 0.130·25-s − 3.20·26-s + 0.181·27-s + 3.63·28-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
| \(L(1)\) |
\(\approx\) |
\(4.354483146\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.354483146\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.46T + 2T^{2} \) |
| 3 | \( 1 + 0.157T + 3T^{2} \) |
| 5 | \( 1 - 2.37T + 5T^{2} \) |
| 7 | \( 1 - 4.70T + 7T^{2} \) |
| 11 | \( 1 + 3.89T + 11T^{2} \) |
| 13 | \( 1 + 6.62T + 13T^{2} \) |
| 17 | \( 1 + 5.95T + 17T^{2} \) |
| 19 | \( 1 + 0.395T + 19T^{2} \) |
| 23 | \( 1 + 1.84T + 23T^{2} \) |
| 29 | \( 1 - 10.6T + 29T^{2} \) |
| 31 | \( 1 - 7.56T + 31T^{2} \) |
| 37 | \( 1 + 5.85T + 37T^{2} \) |
| 41 | \( 1 - 1.13T + 41T^{2} \) |
| 43 | \( 1 - 3.18T + 43T^{2} \) |
| 47 | \( 1 - 0.625T + 47T^{2} \) |
| 53 | \( 1 - 12.8T + 53T^{2} \) |
| 59 | \( 1 + 0.141T + 59T^{2} \) |
| 61 | \( 1 + 3.91T + 61T^{2} \) |
| 67 | \( 1 - 0.202T + 67T^{2} \) |
| 71 | \( 1 - 6.99T + 71T^{2} \) |
| 73 | \( 1 - 0.952T + 73T^{2} \) |
| 79 | \( 1 + 8.10T + 79T^{2} \) |
| 83 | \( 1 - 9.46T + 83T^{2} \) |
| 89 | \( 1 - 10.3T + 89T^{2} \) |
| 97 | \( 1 + 12.2T + 97T^{2} \) |
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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
−10.83785138095375222940113470499, −10.21024098216866125196838700550, −8.697635968721919110704798311913, −7.77124314844195959435707590156, −6.67870697437598678853682679275, −5.65633045469255006571933422940, −4.97169719647278394634446627885, −4.57741022369577636987548766878, −2.48295386106412492564925226448, −2.33060413800315170043219839925,
2.33060413800315170043219839925, 2.48295386106412492564925226448, 4.57741022369577636987548766878, 4.97169719647278394634446627885, 5.65633045469255006571933422940, 6.67870697437598678853682679275, 7.77124314844195959435707590156, 8.697635968721919110704798311913, 10.21024098216866125196838700550, 10.83785138095375222940113470499
