| L(s) = 1 | + 2.56·3-s + 5-s + 5.12·7-s + 3.56·9-s − 4·11-s − 0.561·13-s + 2.56·15-s − 3.12·17-s + 4·19-s + 13.1·21-s + 23-s + 25-s + 1.43·27-s − 8.56·29-s + 1.43·31-s − 10.2·33-s + 5.12·35-s − 7.12·37-s − 1.43·39-s + 0.561·41-s − 9.12·43-s + 3.56·45-s − 3.68·47-s + 19.2·49-s − 8·51-s − 4.24·53-s − 4·55-s + ⋯ |
| L(s) = 1 | + 1.47·3-s + 0.447·5-s + 1.93·7-s + 1.18·9-s − 1.20·11-s − 0.155·13-s + 0.661·15-s − 0.757·17-s + 0.917·19-s + 2.86·21-s + 0.208·23-s + 0.200·25-s + 0.276·27-s − 1.58·29-s + 0.258·31-s − 1.78·33-s + 0.865·35-s − 1.17·37-s − 0.230·39-s + 0.0876·41-s − 1.39·43-s + 0.530·45-s − 0.537·47-s + 2.74·49-s − 1.12·51-s − 0.583·53-s − 0.539·55-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
| \(L(1)\) |
\(\approx\) |
\(3.099131701\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.099131701\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.56T + 3T^{2} \) |
| 7 | \( 1 - 5.12T + 7T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 + 0.561T + 13T^{2} \) |
| 17 | \( 1 + 3.12T + 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 29 | \( 1 + 8.56T + 29T^{2} \) |
| 31 | \( 1 - 1.43T + 31T^{2} \) |
| 37 | \( 1 + 7.12T + 37T^{2} \) |
| 41 | \( 1 - 0.561T + 41T^{2} \) |
| 43 | \( 1 + 9.12T + 43T^{2} \) |
| 47 | \( 1 + 3.68T + 47T^{2} \) |
| 53 | \( 1 + 4.24T + 53T^{2} \) |
| 59 | \( 1 + 6.24T + 59T^{2} \) |
| 61 | \( 1 - 11.1T + 61T^{2} \) |
| 67 | \( 1 - 6.24T + 67T^{2} \) |
| 71 | \( 1 - 3.68T + 71T^{2} \) |
| 73 | \( 1 - 16.5T + 73T^{2} \) |
| 79 | \( 1 + 10.2T + 79T^{2} \) |
| 83 | \( 1 - 12T + 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 + 16.2T + 97T^{2} \) |
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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
−9.920076549679180655210937480297, −9.084272153835367855041594782253, −8.282337019996346633165519184144, −7.86840105021195092401430643316, −7.06434875404388674376261168701, −5.38978448874195467279091161608, −4.84759225581334813187804428545, −3.57529344367571280010444902714, −2.38414527944407314457657815859, −1.71934152560981742499742110002,
1.71934152560981742499742110002, 2.38414527944407314457657815859, 3.57529344367571280010444902714, 4.84759225581334813187804428545, 5.38978448874195467279091161608, 7.06434875404388674376261168701, 7.86840105021195092401430643316, 8.282337019996346633165519184144, 9.084272153835367855041594782253, 9.920076549679180655210937480297
