| L(s) = 1 | + 1.46·3-s + 5-s + 1.53·7-s − 0.860·9-s − 0.860·11-s + 0.139·13-s + 1.46·15-s + 5.50·17-s + 5.25·19-s + 2.24·21-s − 23-s + 25-s − 5.64·27-s + 9.76·29-s + 6.78·31-s − 1.25·33-s + 1.53·35-s − 12.0·37-s + 0.203·39-s − 9.98·41-s + 11.4·43-s − 0.860·45-s − 2.32·47-s − 4.63·49-s + 8.05·51-s + 0.149·53-s − 0.860·55-s + ⋯ |
| L(s) = 1 | + 0.844·3-s + 0.447·5-s + 0.581·7-s − 0.286·9-s − 0.259·11-s + 0.0386·13-s + 0.377·15-s + 1.33·17-s + 1.20·19-s + 0.490·21-s − 0.208·23-s + 0.200·25-s − 1.08·27-s + 1.81·29-s + 1.21·31-s − 0.219·33-s + 0.259·35-s − 1.98·37-s + 0.0325·39-s − 1.55·41-s + 1.74·43-s − 0.128·45-s − 0.338·47-s − 0.662·49-s + 1.12·51-s + 0.0205·53-s − 0.116·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.391957170\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.391957170\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 - 1.46T + 3T^{2} \) |
| 7 | \( 1 - 1.53T + 7T^{2} \) |
| 11 | \( 1 + 0.860T + 11T^{2} \) |
| 13 | \( 1 - 0.139T + 13T^{2} \) |
| 17 | \( 1 - 5.50T + 17T^{2} \) |
| 19 | \( 1 - 5.25T + 19T^{2} \) |
| 29 | \( 1 - 9.76T + 29T^{2} \) |
| 31 | \( 1 - 6.78T + 31T^{2} \) |
| 37 | \( 1 + 12.0T + 37T^{2} \) |
| 41 | \( 1 + 9.98T + 41T^{2} \) |
| 43 | \( 1 - 11.4T + 43T^{2} \) |
| 47 | \( 1 + 2.32T + 47T^{2} \) |
| 53 | \( 1 - 0.149T + 53T^{2} \) |
| 59 | \( 1 + 11.0T + 59T^{2} \) |
| 61 | \( 1 - 4.43T + 61T^{2} \) |
| 67 | \( 1 + 10.4T + 67T^{2} \) |
| 71 | \( 1 - 7.31T + 71T^{2} \) |
| 73 | \( 1 - 7.11T + 73T^{2} \) |
| 79 | \( 1 - 6.79T + 79T^{2} \) |
| 83 | \( 1 - 10.6T + 83T^{2} \) |
| 89 | \( 1 + 17.2T + 89T^{2} \) |
| 97 | \( 1 - 1.93T + 97T^{2} \) |
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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
−9.989798929328177774496466814432, −9.230886428327401376739514601706, −8.263225076565103194560007740773, −7.88629434486098892833799473422, −6.74382005392414375877452518795, −5.60806728842974548021111471315, −4.88392995796946947479376817620, −3.45116035649604447973039219012, −2.69527865241564531053553556130, −1.36312686375066251947537413716,
1.36312686375066251947537413716, 2.69527865241564531053553556130, 3.45116035649604447973039219012, 4.88392995796946947479376817620, 5.60806728842974548021111471315, 6.74382005392414375877452518795, 7.88629434486098892833799473422, 8.263225076565103194560007740773, 9.230886428327401376739514601706, 9.989798929328177774496466814432
