L(s) = 1 | + 0.271·2-s − 1.01·3-s − 1.92·4-s + 4.11·5-s − 0.276·6-s + 2.70·7-s − 1.06·8-s − 1.96·9-s + 1.11·10-s + 2.02·11-s + 1.96·12-s + 1.50·13-s + 0.736·14-s − 4.19·15-s + 3.56·16-s − 3.18·17-s − 0.533·18-s − 6.45·19-s − 7.93·20-s − 2.75·21-s + 0.550·22-s + 8.15·23-s + 1.08·24-s + 11.9·25-s + 0.408·26-s + 5.05·27-s − 5.21·28-s + ⋯ |
L(s) = 1 | + 0.192·2-s − 0.587·3-s − 0.963·4-s + 1.84·5-s − 0.112·6-s + 1.02·7-s − 0.377·8-s − 0.654·9-s + 0.354·10-s + 0.611·11-s + 0.565·12-s + 0.416·13-s + 0.196·14-s − 1.08·15-s + 0.890·16-s − 0.772·17-s − 0.125·18-s − 1.48·19-s − 1.77·20-s − 0.601·21-s + 0.117·22-s + 1.70·23-s + 0.221·24-s + 2.39·25-s + 0.0800·26-s + 0.972·27-s − 0.986·28-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
\(L(1)\) |
\(\approx\) |
\(1.596960220\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.596960220\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.271T + 2T^{2} \) |
| 3 | \( 1 + 1.01T + 3T^{2} \) |
| 5 | \( 1 - 4.11T + 5T^{2} \) |
| 7 | \( 1 - 2.70T + 7T^{2} \) |
| 11 | \( 1 - 2.02T + 11T^{2} \) |
| 13 | \( 1 - 1.50T + 13T^{2} \) |
| 17 | \( 1 + 3.18T + 17T^{2} \) |
| 19 | \( 1 + 6.45T + 19T^{2} \) |
| 23 | \( 1 - 8.15T + 23T^{2} \) |
| 29 | \( 1 - 2.94T + 29T^{2} \) |
| 31 | \( 1 + 0.223T + 31T^{2} \) |
| 37 | \( 1 - 11.8T + 37T^{2} \) |
| 41 | \( 1 + 0.427T + 41T^{2} \) |
| 43 | \( 1 - 2.47T + 43T^{2} \) |
| 47 | \( 1 + 4.43T + 47T^{2} \) |
| 53 | \( 1 - 1.65T + 53T^{2} \) |
| 59 | \( 1 + 8.46T + 59T^{2} \) |
| 61 | \( 1 - 5.39T + 61T^{2} \) |
| 67 | \( 1 + 5.24T + 67T^{2} \) |
| 71 | \( 1 + 3.81T + 71T^{2} \) |
| 73 | \( 1 - 13.1T + 73T^{2} \) |
| 79 | \( 1 - 3.12T + 79T^{2} \) |
| 83 | \( 1 + 17.6T + 83T^{2} \) |
| 89 | \( 1 - 4.40T + 89T^{2} \) |
| 97 | \( 1 + 8.58T + 97T^{2} \) |
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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
−10.77686417498565245118130121982, −9.649924185242280551670206714357, −8.918069491471507616199210760822, −8.381837399833864685056135841890, −6.59123288867519418213676628743, −5.98892762058509875800544983118, −5.11637006761272242291192523007, −4.45048628573988412554217375393, −2.63927170890810956394160981036, −1.24246628498784892666841127290,
1.24246628498784892666841127290, 2.63927170890810956394160981036, 4.45048628573988412554217375393, 5.11637006761272242291192523007, 5.98892762058509875800544983118, 6.59123288867519418213676628743, 8.381837399833864685056135841890, 8.918069491471507616199210760822, 9.649924185242280551670206714357, 10.77686417498565245118130121982