| L(s) = 1 | − 2-s + 2.12·3-s + 4-s + 1.08·5-s − 2.12·6-s + 7-s − 8-s + 1.53·9-s − 1.08·10-s + 3.84·11-s + 2.12·12-s − 0.692·13-s − 14-s + 2.30·15-s + 16-s + 4.21·17-s − 1.53·18-s + 1.86·19-s + 1.08·20-s + 2.12·21-s − 3.84·22-s − 0.145·23-s − 2.12·24-s − 3.82·25-s + 0.692·26-s − 3.12·27-s + 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.22·3-s + 0.5·4-s + 0.484·5-s − 0.869·6-s + 0.377·7-s − 0.353·8-s + 0.510·9-s − 0.342·10-s + 1.16·11-s + 0.614·12-s − 0.192·13-s − 0.267·14-s + 0.595·15-s + 0.250·16-s + 1.02·17-s − 0.361·18-s + 0.428·19-s + 0.242·20-s + 0.464·21-s − 0.820·22-s − 0.0304·23-s − 0.434·24-s − 0.765·25-s + 0.135·26-s − 0.601·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.923674493\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.923674493\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 431 | \( 1 + T \) |
| good | 3 | \( 1 - 2.12T + 3T^{2} \) |
| 5 | \( 1 - 1.08T + 5T^{2} \) |
| 11 | \( 1 - 3.84T + 11T^{2} \) |
| 13 | \( 1 + 0.692T + 13T^{2} \) |
| 17 | \( 1 - 4.21T + 17T^{2} \) |
| 19 | \( 1 - 1.86T + 19T^{2} \) |
| 23 | \( 1 + 0.145T + 23T^{2} \) |
| 29 | \( 1 + 7.24T + 29T^{2} \) |
| 31 | \( 1 + 6.37T + 31T^{2} \) |
| 37 | \( 1 - 11.2T + 37T^{2} \) |
| 41 | \( 1 - 1.15T + 41T^{2} \) |
| 43 | \( 1 - 2.84T + 43T^{2} \) |
| 47 | \( 1 - 8.57T + 47T^{2} \) |
| 53 | \( 1 - 8.48T + 53T^{2} \) |
| 59 | \( 1 - 2.81T + 59T^{2} \) |
| 61 | \( 1 - 9.94T + 61T^{2} \) |
| 67 | \( 1 - 8.12T + 67T^{2} \) |
| 71 | \( 1 + 7.11T + 71T^{2} \) |
| 73 | \( 1 - 1.66T + 73T^{2} \) |
| 79 | \( 1 - 8.37T + 79T^{2} \) |
| 83 | \( 1 - 7.42T + 83T^{2} \) |
| 89 | \( 1 + 16.6T + 89T^{2} \) |
| 97 | \( 1 - 12.4T + 97T^{2} \) |
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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
−8.061784037676570890447114492897, −7.59913146596114770262557429103, −7.00299700108827906638713477797, −5.91034457328394463247591772771, −5.47419331051146005239476691559, −4.04525345182874586999324553699, −3.60015655490460420305184560577, −2.53028425749045960354731339099, −1.92036024579423313015258856881, −0.988303573426780221899858067870,
0.988303573426780221899858067870, 1.92036024579423313015258856881, 2.53028425749045960354731339099, 3.60015655490460420305184560577, 4.04525345182874586999324553699, 5.47419331051146005239476691559, 5.91034457328394463247591772771, 7.00299700108827906638713477797, 7.59913146596114770262557429103, 8.061784037676570890447114492897
