| L(s) = 1 | − 1.37·2-s − 0.725·3-s − 0.121·4-s − 0.225·5-s + 0.994·6-s + 2.47·7-s + 2.90·8-s − 2.47·9-s + 0.308·10-s − 4.06·11-s + 0.0882·12-s + 2.26·13-s − 3.38·14-s + 0.163·15-s − 3.74·16-s + 7.19·17-s + 3.38·18-s − 7.47·19-s + 0.0273·20-s − 1.79·21-s + 5.56·22-s − 0.178·23-s − 2.11·24-s − 4.94·25-s − 3.10·26-s + 3.97·27-s − 0.300·28-s + ⋯ |
| L(s) = 1 | − 0.969·2-s − 0.419·3-s − 0.0607·4-s − 0.100·5-s + 0.406·6-s + 0.934·7-s + 1.02·8-s − 0.824·9-s + 0.0975·10-s − 1.22·11-s + 0.0254·12-s + 0.628·13-s − 0.905·14-s + 0.0421·15-s − 0.935·16-s + 1.74·17-s + 0.798·18-s − 1.71·19-s + 0.00611·20-s − 0.391·21-s + 1.18·22-s − 0.0371·23-s − 0.430·24-s − 0.989·25-s − 0.608·26-s + 0.764·27-s − 0.0568·28-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
| \(L(1)\) |
\(\approx\) |
\(0.6458222760\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6458222760\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.37T + 2T^{2} \) |
| 3 | \( 1 + 0.725T + 3T^{2} \) |
| 5 | \( 1 + 0.225T + 5T^{2} \) |
| 7 | \( 1 - 2.47T + 7T^{2} \) |
| 11 | \( 1 + 4.06T + 11T^{2} \) |
| 13 | \( 1 - 2.26T + 13T^{2} \) |
| 17 | \( 1 - 7.19T + 17T^{2} \) |
| 19 | \( 1 + 7.47T + 19T^{2} \) |
| 23 | \( 1 + 0.178T + 23T^{2} \) |
| 29 | \( 1 - 10.5T + 29T^{2} \) |
| 31 | \( 1 - 1.27T + 31T^{2} \) |
| 37 | \( 1 - 2.08T + 37T^{2} \) |
| 41 | \( 1 - 4.23T + 41T^{2} \) |
| 43 | \( 1 - 8.52T + 43T^{2} \) |
| 47 | \( 1 - 9.96T + 47T^{2} \) |
| 53 | \( 1 - 0.619T + 53T^{2} \) |
| 59 | \( 1 - 11.8T + 59T^{2} \) |
| 61 | \( 1 - 6.30T + 61T^{2} \) |
| 67 | \( 1 - 12.5T + 67T^{2} \) |
| 71 | \( 1 + 9.50T + 71T^{2} \) |
| 73 | \( 1 - 5.14T + 73T^{2} \) |
| 79 | \( 1 + 16.3T + 79T^{2} \) |
| 83 | \( 1 - 4.79T + 83T^{2} \) |
| 89 | \( 1 + 1.64T + 89T^{2} \) |
| 97 | \( 1 - 3.34T + 97T^{2} \) |
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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
−10.56472351021641540393461806373, −9.904341327369648925018167917695, −8.487098956041875842499890100594, −8.321160047047178620052345431506, −7.47712218711359426710170030675, −6.02303477723435028528571373735, −5.19785926794801550532565994363, −4.16414165644554453334526805049, −2.45443668544270497767264085633, −0.834929733689204520348011313208,
0.834929733689204520348011313208, 2.45443668544270497767264085633, 4.16414165644554453334526805049, 5.19785926794801550532565994363, 6.02303477723435028528571373735, 7.47712218711359426710170030675, 8.321160047047178620052345431506, 8.487098956041875842499890100594, 9.904341327369648925018167917695, 10.56472351021641540393461806373
