| L(s) = 1 | + 2.34·2-s + 0.501·3-s + 3.50·4-s + 2.53·5-s + 1.17·6-s − 0.903·7-s + 3.53·8-s − 2.74·9-s + 5.95·10-s − 1.96·11-s + 1.76·12-s + 3.76·13-s − 2.12·14-s + 1.27·15-s + 1.29·16-s + 4.17·17-s − 6.44·18-s − 3.60·19-s + 8.89·20-s − 0.453·21-s − 4.62·22-s − 0.374·23-s + 1.77·24-s + 1.42·25-s + 8.83·26-s − 2.88·27-s − 3.17·28-s + ⋯ |
| L(s) = 1 | + 1.65·2-s + 0.289·3-s + 1.75·4-s + 1.13·5-s + 0.480·6-s − 0.341·7-s + 1.25·8-s − 0.916·9-s + 1.88·10-s − 0.593·11-s + 0.508·12-s + 1.04·13-s − 0.566·14-s + 0.328·15-s + 0.322·16-s + 1.01·17-s − 1.52·18-s − 0.826·19-s + 1.98·20-s − 0.0990·21-s − 0.985·22-s − 0.0781·23-s + 0.362·24-s + 0.285·25-s + 1.73·26-s − 0.555·27-s − 0.599·28-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
| \(L(1)\) |
\(\approx\) |
\(4.256316360\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.256316360\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.34T + 2T^{2} \) |
| 3 | \( 1 - 0.501T + 3T^{2} \) |
| 5 | \( 1 - 2.53T + 5T^{2} \) |
| 7 | \( 1 + 0.903T + 7T^{2} \) |
| 11 | \( 1 + 1.96T + 11T^{2} \) |
| 13 | \( 1 - 3.76T + 13T^{2} \) |
| 17 | \( 1 - 4.17T + 17T^{2} \) |
| 19 | \( 1 + 3.60T + 19T^{2} \) |
| 23 | \( 1 + 0.374T + 23T^{2} \) |
| 29 | \( 1 + 7.23T + 29T^{2} \) |
| 31 | \( 1 + 0.918T + 31T^{2} \) |
| 37 | \( 1 - 6.03T + 37T^{2} \) |
| 41 | \( 1 - 4.61T + 41T^{2} \) |
| 43 | \( 1 + 2.12T + 43T^{2} \) |
| 47 | \( 1 + 4.26T + 47T^{2} \) |
| 53 | \( 1 - 5.29T + 53T^{2} \) |
| 59 | \( 1 - 8.26T + 59T^{2} \) |
| 61 | \( 1 - 5.99T + 61T^{2} \) |
| 67 | \( 1 + 1.02T + 67T^{2} \) |
| 71 | \( 1 + 11.8T + 71T^{2} \) |
| 73 | \( 1 - 3.45T + 73T^{2} \) |
| 79 | \( 1 + 1.04T + 79T^{2} \) |
| 83 | \( 1 - 12.6T + 83T^{2} \) |
| 89 | \( 1 + 0.804T + 89T^{2} \) |
| 97 | \( 1 - 1.25T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.89762921368632536836651605049, −9.883451304809993540862273209700, −8.918535585326032995775629117286, −7.82158688269244044757088596299, −6.48308868492573180295957835399, −5.82594768169620701154601726972, −5.33442396452248341040607924111, −3.92524286248829784312358522282, −3.01347562557292393802384484967, −2.04574174974082331927372056624,
2.04574174974082331927372056624, 3.01347562557292393802384484967, 3.92524286248829784312358522282, 5.33442396452248341040607924111, 5.82594768169620701154601726972, 6.48308868492573180295957835399, 7.82158688269244044757088596299, 8.918535585326032995775629117286, 9.883451304809993540862273209700, 10.89762921368632536836651605049
