| L(s) = 1 | − 2.96·3-s + 5-s − 1.02·7-s + 5.79·9-s + 0.504·13-s − 2.96·15-s + 3.93·17-s − 2.50·19-s + 3.05·21-s − 5.73·23-s + 25-s − 8.29·27-s − 6.92·29-s + 4.47·31-s − 1.02·35-s + 4.78·37-s − 1.49·39-s − 11.8·41-s + 7.10·43-s + 5.79·45-s − 0.182·47-s − 5.94·49-s − 11.6·51-s + 12.4·53-s + 7.42·57-s − 10.5·59-s + 10.9·61-s + ⋯ |
| L(s) = 1 | − 1.71·3-s + 0.447·5-s − 0.388·7-s + 1.93·9-s + 0.139·13-s − 0.765·15-s + 0.953·17-s − 0.574·19-s + 0.665·21-s − 1.19·23-s + 0.200·25-s − 1.59·27-s − 1.28·29-s + 0.804·31-s − 0.173·35-s + 0.786·37-s − 0.239·39-s − 1.84·41-s + 1.08·43-s + 0.864·45-s − 0.0266·47-s − 0.848·49-s − 1.63·51-s + 1.71·53-s + 0.983·57-s − 1.37·59-s + 1.39·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + 2.96T + 3T^{2} \) |
| 7 | \( 1 + 1.02T + 7T^{2} \) |
| 13 | \( 1 - 0.504T + 13T^{2} \) |
| 17 | \( 1 - 3.93T + 17T^{2} \) |
| 19 | \( 1 + 2.50T + 19T^{2} \) |
| 23 | \( 1 + 5.73T + 23T^{2} \) |
| 29 | \( 1 + 6.92T + 29T^{2} \) |
| 31 | \( 1 - 4.47T + 31T^{2} \) |
| 37 | \( 1 - 4.78T + 37T^{2} \) |
| 41 | \( 1 + 11.8T + 41T^{2} \) |
| 43 | \( 1 - 7.10T + 43T^{2} \) |
| 47 | \( 1 + 0.182T + 47T^{2} \) |
| 53 | \( 1 - 12.4T + 53T^{2} \) |
| 59 | \( 1 + 10.5T + 59T^{2} \) |
| 61 | \( 1 - 10.9T + 61T^{2} \) |
| 67 | \( 1 - 9.76T + 67T^{2} \) |
| 71 | \( 1 + 13.9T + 71T^{2} \) |
| 73 | \( 1 - 7.32T + 73T^{2} \) |
| 79 | \( 1 + 10.2T + 79T^{2} \) |
| 83 | \( 1 + 4.33T + 83T^{2} \) |
| 89 | \( 1 - 17.2T + 89T^{2} \) |
| 97 | \( 1 + 7.80T + 97T^{2} \) |
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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
−7.72125664405812884844855176481, −6.96460625047188400819141370397, −6.22397639973494794760563929532, −5.83957050369178062465101398347, −5.18584766556543072331711453674, −4.35028037059269154736930087422, −3.52062240393997614157804764761, −2.17253712569091146842021731432, −1.12300771710951077959255575604, 0,
1.12300771710951077959255575604, 2.17253712569091146842021731432, 3.52062240393997614157804764761, 4.35028037059269154736930087422, 5.18584766556543072331711453674, 5.83957050369178062465101398347, 6.22397639973494794760563929532, 6.96460625047188400819141370397, 7.72125664405812884844855176481
