| L(s) = 1 | − 32·2-s + 243·3-s + 1.02e3·4-s − 1.17e4·5-s − 7.77e3·6-s − 5.00e4·7-s − 3.27e4·8-s + 5.90e4·9-s + 3.75e5·10-s − 5.31e5·11-s + 2.48e5·12-s + 1.33e6·13-s + 1.60e6·14-s − 2.85e6·15-s + 1.04e6·16-s − 5.10e6·17-s − 1.88e6·18-s + 2.90e6·19-s − 1.20e7·20-s − 1.21e7·21-s + 1.70e7·22-s + 3.05e7·23-s − 7.96e6·24-s + 8.87e7·25-s − 4.26e7·26-s + 1.43e7·27-s − 5.12e7·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.67·5-s − 0.408·6-s − 1.12·7-s − 0.353·8-s + 1/3·9-s + 1.18·10-s − 0.994·11-s + 0.288·12-s + 0.995·13-s + 0.795·14-s − 0.969·15-s + 1/4·16-s − 0.872·17-s − 0.235·18-s + 0.268·19-s − 0.839·20-s − 0.649·21-s + 0.703·22-s + 0.991·23-s − 0.204·24-s + 1.81·25-s − 0.703·26-s + 0.192·27-s − 0.562·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{5} T \) |
| 3 | \( 1 - p^{5} T \) |
| good | 5 | \( 1 + 2346 p T + p^{11} T^{2} \) |
| 7 | \( 1 + 7144 p T + p^{11} T^{2} \) |
| 11 | \( 1 + 531420 T + p^{11} T^{2} \) |
| 13 | \( 1 - 1332566 T + p^{11} T^{2} \) |
| 17 | \( 1 + 5109678 T + p^{11} T^{2} \) |
| 19 | \( 1 - 2901404 T + p^{11} T^{2} \) |
| 23 | \( 1 - 30597000 T + p^{11} T^{2} \) |
| 29 | \( 1 + 77006634 T + p^{11} T^{2} \) |
| 31 | \( 1 + 239418352 T + p^{11} T^{2} \) |
| 37 | \( 1 + 785041666 T + p^{11} T^{2} \) |
| 41 | \( 1 - 411252954 T + p^{11} T^{2} \) |
| 43 | \( 1 - 351233348 T + p^{11} T^{2} \) |
| 47 | \( 1 - 95821680 T + p^{11} T^{2} \) |
| 53 | \( 1 + 1465857378 T + p^{11} T^{2} \) |
| 59 | \( 1 - 5621152020 T + p^{11} T^{2} \) |
| 61 | \( 1 + 10473587770 T + p^{11} T^{2} \) |
| 67 | \( 1 - 4515307532 T + p^{11} T^{2} \) |
| 71 | \( 1 + 8509579560 T + p^{11} T^{2} \) |
| 73 | \( 1 - 2012496986 T + p^{11} T^{2} \) |
| 79 | \( 1 + 22238409568 T + p^{11} T^{2} \) |
| 83 | \( 1 - 6328647516 T + p^{11} T^{2} \) |
| 89 | \( 1 + 50123706678 T + p^{11} T^{2} \) |
| 97 | \( 1 - 94805961314 T + p^{11} T^{2} \) |
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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
−19.49516154158430969660939025316, −18.58779193288217414261624772106, −16.12573529230515993637501845118, −15.43605882262235197027193221341, −12.86772280091318974721978484808, −10.95046201592138460298285860555, −8.812153641762323252267694640716, −7.29104514388497780703688162385, −3.40042358428150328610555996178, 0,
3.40042358428150328610555996178, 7.29104514388497780703688162385, 8.812153641762323252267694640716, 10.95046201592138460298285860555, 12.86772280091318974721978484808, 15.43605882262235197027193221341, 16.12573529230515993637501845118, 18.58779193288217414261624772106, 19.49516154158430969660939025316
