| L(s) = 1 | − 0.656·2-s − 1.56·4-s − 3.35·5-s − 2.47·7-s + 2.34·8-s + 2.20·10-s + 1.78·11-s + 0.0701·13-s + 1.62·14-s + 1.59·16-s − 6.97·17-s − 5.06·19-s + 5.25·20-s − 1.17·22-s + 23-s + 6.24·25-s − 0.0461·26-s + 3.88·28-s + 29-s − 4.46·31-s − 5.73·32-s + 4.58·34-s + 8.31·35-s − 9.00·37-s + 3.32·38-s − 7.86·40-s − 0.277·41-s + ⋯ |
| L(s) = 1 | − 0.464·2-s − 0.784·4-s − 1.49·5-s − 0.937·7-s + 0.828·8-s + 0.696·10-s + 0.539·11-s + 0.0194·13-s + 0.435·14-s + 0.399·16-s − 1.69·17-s − 1.16·19-s + 1.17·20-s − 0.250·22-s + 0.208·23-s + 1.24·25-s − 0.00904·26-s + 0.735·28-s + 0.185·29-s − 0.801·31-s − 1.01·32-s + 0.785·34-s + 1.40·35-s − 1.47·37-s + 0.539·38-s − 1.24·40-s − 0.0433·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.01141601931\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01141601931\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 + 0.656T + 2T^{2} \) |
| 5 | \( 1 + 3.35T + 5T^{2} \) |
| 7 | \( 1 + 2.47T + 7T^{2} \) |
| 11 | \( 1 - 1.78T + 11T^{2} \) |
| 13 | \( 1 - 0.0701T + 13T^{2} \) |
| 17 | \( 1 + 6.97T + 17T^{2} \) |
| 19 | \( 1 + 5.06T + 19T^{2} \) |
| 31 | \( 1 + 4.46T + 31T^{2} \) |
| 37 | \( 1 + 9.00T + 37T^{2} \) |
| 41 | \( 1 + 0.277T + 41T^{2} \) |
| 43 | \( 1 + 7.62T + 43T^{2} \) |
| 47 | \( 1 - 3.83T + 47T^{2} \) |
| 53 | \( 1 + 3.59T + 53T^{2} \) |
| 59 | \( 1 - 4.41T + 59T^{2} \) |
| 61 | \( 1 + 2.15T + 61T^{2} \) |
| 67 | \( 1 + 7.48T + 67T^{2} \) |
| 71 | \( 1 + 13.0T + 71T^{2} \) |
| 73 | \( 1 + 5.44T + 73T^{2} \) |
| 79 | \( 1 + 14.3T + 79T^{2} \) |
| 83 | \( 1 + 12.6T + 83T^{2} \) |
| 89 | \( 1 - 2.48T + 89T^{2} \) |
| 97 | \( 1 + 11.2T + 97T^{2} \) |
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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
−8.334184363715864581801775537483, −7.33114150487846917792875811950, −6.91611155997138571705997621509, −6.12841619987171006187031074113, −4.95520319180485749660443699021, −4.24254084957392298598584909468, −3.85460198953159095052204511219, −2.98529444985884052605829530439, −1.61984698257582156379821866742, −0.06037015171813126189267670696,
0.06037015171813126189267670696, 1.61984698257582156379821866742, 2.98529444985884052605829530439, 3.85460198953159095052204511219, 4.24254084957392298598584909468, 4.95520319180485749660443699021, 6.12841619987171006187031074113, 6.91611155997138571705997621509, 7.33114150487846917792875811950, 8.334184363715864581801775537483
