| L(s) = 1 | − 2.59·2-s − 25.2·4-s + 1.42·5-s + 160.·7-s + 148.·8-s − 3.69·10-s − 121·11-s + 370.·13-s − 415.·14-s + 422.·16-s + 297.·17-s − 1.91e3·19-s − 35.9·20-s + 314.·22-s − 32.4·23-s − 3.12e3·25-s − 962.·26-s − 4.04e3·28-s + 5.50e3·29-s − 6.82e3·31-s − 5.85e3·32-s − 771.·34-s + 227.·35-s + 7.50e3·37-s + 4.98e3·38-s + 211.·40-s − 1.60e4·41-s + ⋯ |
| L(s) = 1 | − 0.459·2-s − 0.789·4-s + 0.0254·5-s + 1.23·7-s + 0.821·8-s − 0.0116·10-s − 0.301·11-s + 0.608·13-s − 0.566·14-s + 0.412·16-s + 0.249·17-s − 1.21·19-s − 0.0200·20-s + 0.138·22-s − 0.0127·23-s − 0.999·25-s − 0.279·26-s − 0.974·28-s + 1.21·29-s − 1.27·31-s − 1.01·32-s − 0.114·34-s + 0.0314·35-s + 0.901·37-s + 0.560·38-s + 0.0209·40-s − 1.49·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 11 | \( 1 + 121T \) |
| good | 2 | \( 1 + 2.59T + 32T^{2} \) |
| 5 | \( 1 - 1.42T + 3.12e3T^{2} \) |
| 7 | \( 1 - 160.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 370.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 297.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.91e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 32.4T + 6.43e6T^{2} \) |
| 29 | \( 1 - 5.50e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 6.82e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 7.50e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.60e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.82e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.19e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 200.T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.57e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.45e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.86e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 4.88e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.21e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 5.23e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.22e5T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.34e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.45e5T + 8.58e9T^{2} \) |
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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
−8.826785492642870110828098150117, −8.174479938188225961858608292442, −7.67761242452350632595866467286, −6.35931197991415324479256140927, −5.28978838015374587296509002747, −4.55672143617278371235125226917, −3.68702676337987782651352664788, −2.09495951098310649550185215757, −1.17973534609123647411743026058, 0,
1.17973534609123647411743026058, 2.09495951098310649550185215757, 3.68702676337987782651352664788, 4.55672143617278371235125226917, 5.28978838015374587296509002747, 6.35931197991415324479256140927, 7.67761242452350632595866467286, 8.174479938188225961858608292442, 8.826785492642870110828098150117
