| L(s) = 1 | − 0.386·2-s − 1.55·3-s − 1.85·4-s − 3.75·5-s + 0.600·6-s − 2.84·7-s + 1.48·8-s − 0.587·9-s + 1.45·10-s + 0.341·11-s + 2.87·12-s − 4.58·13-s + 1.10·14-s + 5.82·15-s + 3.12·16-s − 4.50·17-s + 0.227·18-s − 4.60·19-s + 6.94·20-s + 4.42·21-s − 0.131·22-s − 6.46·23-s − 2.31·24-s + 9.06·25-s + 1.77·26-s + 5.57·27-s + 5.26·28-s + ⋯ |
| L(s) = 1 | − 0.273·2-s − 0.896·3-s − 0.925·4-s − 1.67·5-s + 0.245·6-s − 1.07·7-s + 0.526·8-s − 0.195·9-s + 0.458·10-s + 0.102·11-s + 0.829·12-s − 1.27·13-s + 0.294·14-s + 1.50·15-s + 0.781·16-s − 1.09·17-s + 0.0535·18-s − 1.05·19-s + 1.55·20-s + 0.964·21-s − 0.0281·22-s − 1.34·23-s − 0.472·24-s + 1.81·25-s + 0.347·26-s + 1.07·27-s + 0.995·28-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
| \(L(1)\) |
\(\approx\) |
\(0.06363881254\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.06363881254\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.386T + 2T^{2} \) |
| 3 | \( 1 + 1.55T + 3T^{2} \) |
| 5 | \( 1 + 3.75T + 5T^{2} \) |
| 7 | \( 1 + 2.84T + 7T^{2} \) |
| 11 | \( 1 - 0.341T + 11T^{2} \) |
| 13 | \( 1 + 4.58T + 13T^{2} \) |
| 17 | \( 1 + 4.50T + 17T^{2} \) |
| 19 | \( 1 + 4.60T + 19T^{2} \) |
| 23 | \( 1 + 6.46T + 23T^{2} \) |
| 29 | \( 1 - 9.68T + 29T^{2} \) |
| 31 | \( 1 - 7.65T + 31T^{2} \) |
| 37 | \( 1 + 0.361T + 37T^{2} \) |
| 41 | \( 1 + 7.84T + 41T^{2} \) |
| 43 | \( 1 + 9.06T + 43T^{2} \) |
| 47 | \( 1 - 2.68T + 47T^{2} \) |
| 53 | \( 1 + 1.93T + 53T^{2} \) |
| 59 | \( 1 + 0.518T + 59T^{2} \) |
| 61 | \( 1 + 7.90T + 61T^{2} \) |
| 67 | \( 1 - 6.35T + 67T^{2} \) |
| 71 | \( 1 + 6.34T + 71T^{2} \) |
| 73 | \( 1 + 13.6T + 73T^{2} \) |
| 79 | \( 1 + 12.1T + 79T^{2} \) |
| 83 | \( 1 + 5.17T + 83T^{2} \) |
| 89 | \( 1 - 0.605T + 89T^{2} \) |
| 97 | \( 1 + 0.592T + 97T^{2} \) |
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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
−10.47533059154947439281212610698, −9.964225609694205849680056934217, −8.655561108195490985124446576465, −8.212684810923393971481405377217, −7.00198377498718291331980533926, −6.24724590260545764618968855583, −4.75716598725790353372355829722, −4.32093092285993047116871322779, −3.03342590860552507585939976467, −0.22359698785891014393310962308,
0.22359698785891014393310962308, 3.03342590860552507585939976467, 4.32093092285993047116871322779, 4.75716598725790353372355829722, 6.24724590260545764618968855583, 7.00198377498718291331980533926, 8.212684810923393971481405377217, 8.655561108195490985124446576465, 9.964225609694205849680056934217, 10.47533059154947439281212610698
