| L(s) = 1 | + 0.838·3-s + 2.50·5-s − 7-s − 2.29·9-s + 3.46·11-s + 6.36·13-s + 2.10·15-s + 6.34·17-s + 19-s − 0.838·21-s − 5.29·23-s + 1.29·25-s − 4.44·27-s + 10.1·29-s − 9.39·31-s + 2.90·33-s − 2.50·35-s − 2.14·37-s + 5.33·39-s − 7.43·41-s + 1.15·43-s − 5.76·45-s + 5.96·47-s + 49-s + 5.31·51-s + 6.50·53-s + 8.68·55-s + ⋯ |
| L(s) = 1 | + 0.484·3-s + 1.12·5-s − 0.377·7-s − 0.765·9-s + 1.04·11-s + 1.76·13-s + 0.543·15-s + 1.53·17-s + 0.229·19-s − 0.183·21-s − 1.10·23-s + 0.259·25-s − 0.855·27-s + 1.88·29-s − 1.68·31-s + 0.505·33-s − 0.424·35-s − 0.352·37-s + 0.854·39-s − 1.16·41-s + 0.176·43-s − 0.859·45-s + 0.870·47-s + 0.142·49-s + 0.744·51-s + 0.893·53-s + 1.17·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.560677511\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.560677511\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 0.838T + 3T^{2} \) |
| 5 | \( 1 - 2.50T + 5T^{2} \) |
| 11 | \( 1 - 3.46T + 11T^{2} \) |
| 13 | \( 1 - 6.36T + 13T^{2} \) |
| 17 | \( 1 - 6.34T + 17T^{2} \) |
| 23 | \( 1 + 5.29T + 23T^{2} \) |
| 29 | \( 1 - 10.1T + 29T^{2} \) |
| 31 | \( 1 + 9.39T + 31T^{2} \) |
| 37 | \( 1 + 2.14T + 37T^{2} \) |
| 41 | \( 1 + 7.43T + 41T^{2} \) |
| 43 | \( 1 - 1.15T + 43T^{2} \) |
| 47 | \( 1 - 5.96T + 47T^{2} \) |
| 53 | \( 1 - 6.50T + 53T^{2} \) |
| 59 | \( 1 - 9.26T + 59T^{2} \) |
| 61 | \( 1 - 7.12T + 61T^{2} \) |
| 67 | \( 1 - 7.54T + 67T^{2} \) |
| 71 | \( 1 - 1.10T + 71T^{2} \) |
| 73 | \( 1 - 0.0584T + 73T^{2} \) |
| 79 | \( 1 - 16.5T + 79T^{2} \) |
| 83 | \( 1 + 5.59T + 83T^{2} \) |
| 89 | \( 1 + 14.8T + 89T^{2} \) |
| 97 | \( 1 + 1.89T + 97T^{2} \) |
| 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
−8.014539282293915027244641464743, −6.91590376391268021267707965273, −6.33799744634286180398432221367, −5.70517928225515226328280517172, −5.36680662818867136953934721051, −3.79190676490131727631772285386, −3.66000347614232771499337033994, −2.64780957267899985759443723181, −1.74171990735197728761624661627, −0.964762793471881548466605779987,
0.964762793471881548466605779987, 1.74171990735197728761624661627, 2.64780957267899985759443723181, 3.66000347614232771499337033994, 3.79190676490131727631772285386, 5.36680662818867136953934721051, 5.70517928225515226328280517172, 6.33799744634286180398432221367, 6.91590376391268021267707965273, 8.014539282293915027244641464743
