| L(s) = 1 | + 2.21·3-s + 3.09·5-s + 4.67·7-s + 1.90·9-s + 4.76·11-s − 4.18·13-s + 6.86·15-s − 2.76·17-s + 4.43·19-s + 10.3·21-s − 8.88·23-s + 4.59·25-s − 2.41·27-s + 2.14·29-s − 7.46·31-s + 10.5·33-s + 14.4·35-s + 9.30·37-s − 9.26·39-s − 8.39·41-s + 3.77·43-s + 5.90·45-s + 3.41·47-s + 14.9·49-s − 6.12·51-s + 11.5·53-s + 14.7·55-s + ⋯ |
| L(s) = 1 | + 1.27·3-s + 1.38·5-s + 1.76·7-s + 0.635·9-s + 1.43·11-s − 1.15·13-s + 1.77·15-s − 0.670·17-s + 1.01·19-s + 2.26·21-s − 1.85·23-s + 0.918·25-s − 0.465·27-s + 0.399·29-s − 1.34·31-s + 1.83·33-s + 2.44·35-s + 1.52·37-s − 1.48·39-s − 1.31·41-s + 0.576·43-s + 0.880·45-s + 0.498·47-s + 2.12·49-s − 0.857·51-s + 1.58·53-s + 1.98·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.944312759\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.944312759\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 251 | \( 1 - T \) |
| good | 3 | \( 1 - 2.21T + 3T^{2} \) |
| 5 | \( 1 - 3.09T + 5T^{2} \) |
| 7 | \( 1 - 4.67T + 7T^{2} \) |
| 11 | \( 1 - 4.76T + 11T^{2} \) |
| 13 | \( 1 + 4.18T + 13T^{2} \) |
| 17 | \( 1 + 2.76T + 17T^{2} \) |
| 19 | \( 1 - 4.43T + 19T^{2} \) |
| 23 | \( 1 + 8.88T + 23T^{2} \) |
| 29 | \( 1 - 2.14T + 29T^{2} \) |
| 31 | \( 1 + 7.46T + 31T^{2} \) |
| 37 | \( 1 - 9.30T + 37T^{2} \) |
| 41 | \( 1 + 8.39T + 41T^{2} \) |
| 43 | \( 1 - 3.77T + 43T^{2} \) |
| 47 | \( 1 - 3.41T + 47T^{2} \) |
| 53 | \( 1 - 11.5T + 53T^{2} \) |
| 59 | \( 1 + 0.620T + 59T^{2} \) |
| 61 | \( 1 + 11.0T + 61T^{2} \) |
| 67 | \( 1 + 3.88T + 67T^{2} \) |
| 71 | \( 1 + 5.36T + 71T^{2} \) |
| 73 | \( 1 - 1.41T + 73T^{2} \) |
| 79 | \( 1 + 6.15T + 79T^{2} \) |
| 83 | \( 1 + 4.20T + 83T^{2} \) |
| 89 | \( 1 + 16.8T + 89T^{2} \) |
| 97 | \( 1 - 9.25T + 97T^{2} \) |
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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
−8.692194052824346285441798736820, −7.68606378248441816697934665597, −7.31258024649243782930419194027, −6.14459883107520123108288231423, −5.48179443193174163546726806250, −4.56860823415893187339225861052, −3.89915870015030187047997275902, −2.61924803759791608951568042854, −1.98291349279928036952712806466, −1.45476707413079181268051807104,
1.45476707413079181268051807104, 1.98291349279928036952712806466, 2.61924803759791608951568042854, 3.89915870015030187047997275902, 4.56860823415893187339225861052, 5.48179443193174163546726806250, 6.14459883107520123108288231423, 7.31258024649243782930419194027, 7.68606378248441816697934665597, 8.692194052824346285441798736820
