| L(s) = 1 | − 4.30·2-s − 22.8·3-s − 13.4·4-s + 59.7·5-s + 98.4·6-s + 76.0·7-s + 195.·8-s + 280.·9-s − 257.·10-s − 158.·11-s + 308.·12-s − 69.3·13-s − 327.·14-s − 1.36e3·15-s − 410.·16-s − 1.36e3·17-s − 1.20e3·18-s + 1.26e3·19-s − 805.·20-s − 1.74e3·21-s + 681.·22-s + 294.·23-s − 4.47e3·24-s + 443.·25-s + 298.·26-s − 866.·27-s − 1.02e3·28-s + ⋯ |
| L(s) = 1 | − 0.760·2-s − 1.46·3-s − 0.421·4-s + 1.06·5-s + 1.11·6-s + 0.586·7-s + 1.08·8-s + 1.15·9-s − 0.812·10-s − 0.394·11-s + 0.618·12-s − 0.113·13-s − 0.446·14-s − 1.56·15-s − 0.401·16-s − 1.14·17-s − 0.879·18-s + 0.800·19-s − 0.450·20-s − 0.861·21-s + 0.300·22-s + 0.116·23-s − 1.58·24-s + 0.141·25-s + 0.0865·26-s − 0.228·27-s − 0.247·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 71 | \( 1 + 5.04e3T \) |
| good | 2 | \( 1 + 4.30T + 32T^{2} \) |
| 3 | \( 1 + 22.8T + 243T^{2} \) |
| 5 | \( 1 - 59.7T + 3.12e3T^{2} \) |
| 7 | \( 1 - 76.0T + 1.68e4T^{2} \) |
| 11 | \( 1 + 158.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 69.3T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.36e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.26e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 294.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 3.39e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 3.29e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.31e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.45e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.06e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.92e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.83e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.21e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.25e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.81e3T + 1.35e9T^{2} \) |
| 73 | \( 1 + 3.63e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 8.44e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.01e5T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.62e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 3.87e4T + 8.58e9T^{2} \) |
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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
−13.14679369250696379642404073456, −11.74471167657427463890329394388, −10.68272300416744682628015655001, −9.922965447757561208206775488801, −8.678466012475244219883935700684, −7.08014189409974733207548390603, −5.64321955420953027781387007113, −4.76633616268011897382841376474, −1.58319221071649347455237037522, 0,
1.58319221071649347455237037522, 4.76633616268011897382841376474, 5.64321955420953027781387007113, 7.08014189409974733207548390603, 8.678466012475244219883935700684, 9.922965447757561208206775488801, 10.68272300416744682628015655001, 11.74471167657427463890329394388, 13.14679369250696379642404073456
