| L(s) = 1 | − 2-s + 4-s + 0.460·5-s − 8-s − 0.460·10-s + 3.64·11-s − 1.46·13-s + 16-s − 3.73·17-s + 4.05·19-s + 0.460·20-s − 3.64·22-s − 1.13·23-s − 4.78·25-s + 1.46·26-s − 8.97·29-s − 0.514·31-s − 32-s + 3.73·34-s + 9.10·37-s − 4.05·38-s − 0.460·40-s − 0.945·41-s − 9.32·43-s + 3.64·44-s + 1.13·46-s − 2.32·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.205·5-s − 0.353·8-s − 0.145·10-s + 1.09·11-s − 0.405·13-s + 0.250·16-s − 0.905·17-s + 0.930·19-s + 0.102·20-s − 0.777·22-s − 0.236·23-s − 0.957·25-s + 0.286·26-s − 1.66·29-s − 0.0924·31-s − 0.176·32-s + 0.640·34-s + 1.49·37-s − 0.657·38-s − 0.0728·40-s − 0.147·41-s − 1.42·43-s + 0.549·44-s + 0.167·46-s − 0.339·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7938 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7938 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.389273021\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.389273021\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 0.460T + 5T^{2} \) |
| 11 | \( 1 - 3.64T + 11T^{2} \) |
| 13 | \( 1 + 1.46T + 13T^{2} \) |
| 17 | \( 1 + 3.73T + 17T^{2} \) |
| 19 | \( 1 - 4.05T + 19T^{2} \) |
| 23 | \( 1 + 1.13T + 23T^{2} \) |
| 29 | \( 1 + 8.97T + 29T^{2} \) |
| 31 | \( 1 + 0.514T + 31T^{2} \) |
| 37 | \( 1 - 9.10T + 37T^{2} \) |
| 41 | \( 1 + 0.945T + 41T^{2} \) |
| 43 | \( 1 + 9.32T + 43T^{2} \) |
| 47 | \( 1 + 2.32T + 47T^{2} \) |
| 53 | \( 1 - 12.4T + 53T^{2} \) |
| 59 | \( 1 - 12.8T + 59T^{2} \) |
| 61 | \( 1 - 12.0T + 61T^{2} \) |
| 67 | \( 1 + 2.32T + 67T^{2} \) |
| 71 | \( 1 + 1.67T + 71T^{2} \) |
| 73 | \( 1 - 13.2T + 73T^{2} \) |
| 79 | \( 1 + 5.00T + 79T^{2} \) |
| 83 | \( 1 - 6.64T + 83T^{2} \) |
| 89 | \( 1 + 2.72T + 89T^{2} \) |
| 97 | \( 1 - 11.1T + 97T^{2} \) |
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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
−7.87224912887039963061392550704, −7.14555867065874166831053934084, −6.65551499342719173760923371774, −5.85170487741119167860091728890, −5.20866866356297353246624276490, −4.11979685317136410567481426320, −3.55564967937104701632523887332, −2.38981404392840356890406910780, −1.75560973161449502453483437359, −0.65143967264169732672994957380,
0.65143967264169732672994957380, 1.75560973161449502453483437359, 2.38981404392840356890406910780, 3.55564967937104701632523887332, 4.11979685317136410567481426320, 5.20866866356297353246624276490, 5.85170487741119167860091728890, 6.65551499342719173760923371774, 7.14555867065874166831053934084, 7.87224912887039963061392550704
