| L(s) = 1 | − 2.82·5-s + 3.46·7-s + 11-s − 6.89·13-s + 6.29·17-s − 6.29·19-s − 4.89·23-s + 3.00·25-s + 0.635·29-s − 5.65·31-s − 9.79·35-s + 7.79·37-s − 0.635·41-s + 0.635·43-s + 8.89·47-s + 4.99·49-s + 9.75·53-s − 2.82·55-s + 13.7·59-s − 1.10·61-s + 19.5·65-s − 0.898·71-s + 6·73-s + 3.46·77-s − 16.0·79-s + 13.7·83-s − 17.7·85-s + ⋯ |
| L(s) = 1 | − 1.26·5-s + 1.30·7-s + 0.301·11-s − 1.91·13-s + 1.52·17-s − 1.44·19-s − 1.02·23-s + 0.600·25-s + 0.118·29-s − 1.01·31-s − 1.65·35-s + 1.28·37-s − 0.0992·41-s + 0.0969·43-s + 1.29·47-s + 0.714·49-s + 1.34·53-s − 0.381·55-s + 1.79·59-s − 0.140·61-s + 2.42·65-s − 0.106·71-s + 0.702·73-s + 0.394·77-s − 1.80·79-s + 1.51·83-s − 1.93·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.345895413\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.345895413\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 + 2.82T + 5T^{2} \) |
| 7 | \( 1 - 3.46T + 7T^{2} \) |
| 13 | \( 1 + 6.89T + 13T^{2} \) |
| 17 | \( 1 - 6.29T + 17T^{2} \) |
| 19 | \( 1 + 6.29T + 19T^{2} \) |
| 23 | \( 1 + 4.89T + 23T^{2} \) |
| 29 | \( 1 - 0.635T + 29T^{2} \) |
| 31 | \( 1 + 5.65T + 31T^{2} \) |
| 37 | \( 1 - 7.79T + 37T^{2} \) |
| 41 | \( 1 + 0.635T + 41T^{2} \) |
| 43 | \( 1 - 0.635T + 43T^{2} \) |
| 47 | \( 1 - 8.89T + 47T^{2} \) |
| 53 | \( 1 - 9.75T + 53T^{2} \) |
| 59 | \( 1 - 13.7T + 59T^{2} \) |
| 61 | \( 1 + 1.10T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 0.898T + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 + 16.0T + 79T^{2} \) |
| 83 | \( 1 - 13.7T + 83T^{2} \) |
| 89 | \( 1 + 11.3T + 89T^{2} \) |
| 97 | \( 1 - 10T + 97T^{2} \) |
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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
−7.919355665306829998614285622647, −7.53921853002665644329948470206, −6.90715597475580834098218716982, −5.71897629566622795753605760465, −5.11590496826968088858760888379, −4.21621417447859749562401606764, −3.97677906692697471945510106275, −2.68120004410189837251555530456, −1.89004548597811525240050405604, −0.59399337846080600064488420933,
0.59399337846080600064488420933, 1.89004548597811525240050405604, 2.68120004410189837251555530456, 3.97677906692697471945510106275, 4.21621417447859749562401606764, 5.11590496826968088858760888379, 5.71897629566622795753605760465, 6.90715597475580834098218716982, 7.53921853002665644329948470206, 7.919355665306829998614285622647
