| L(s) = 1 | − 69.6·2-s − 243·3-s + 2.80e3·4-s − 8.53e3·5-s + 1.69e4·6-s + 1.68e4·7-s − 5.28e4·8-s + 5.90e4·9-s + 5.94e5·10-s − 8.96e5·11-s − 6.82e5·12-s − 3.71e5·13-s − 1.17e6·14-s + 2.07e6·15-s − 2.06e6·16-s + 6.97e6·17-s − 4.11e6·18-s + 8.73e6·19-s − 2.39e7·20-s − 4.08e6·21-s + 6.24e7·22-s − 2.99e7·23-s + 1.28e7·24-s + 2.39e7·25-s + 2.58e7·26-s − 1.43e7·27-s + 4.71e7·28-s + ⋯ |
| L(s) = 1 | − 1.53·2-s − 0.577·3-s + 1.37·4-s − 1.22·5-s + 0.888·6-s + 0.377·7-s − 0.570·8-s + 0.333·9-s + 1.88·10-s − 1.67·11-s − 0.791·12-s − 0.277·13-s − 0.581·14-s + 0.705·15-s − 0.491·16-s + 1.19·17-s − 0.513·18-s + 0.809·19-s − 1.67·20-s − 0.218·21-s + 2.58·22-s − 0.971·23-s + 0.329·24-s + 0.491·25-s + 0.427·26-s − 0.192·27-s + 0.518·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + 243T \) |
| 7 | \( 1 - 1.68e4T \) |
| 13 | \( 1 + 3.71e5T \) |
| good | 2 | \( 1 + 69.6T + 2.04e3T^{2} \) |
| 5 | \( 1 + 8.53e3T + 4.88e7T^{2} \) |
| 11 | \( 1 + 8.96e5T + 2.85e11T^{2} \) |
| 17 | \( 1 - 6.97e6T + 3.42e13T^{2} \) |
| 19 | \( 1 - 8.73e6T + 1.16e14T^{2} \) |
| 23 | \( 1 + 2.99e7T + 9.52e14T^{2} \) |
| 29 | \( 1 + 2.58e7T + 1.22e16T^{2} \) |
| 31 | \( 1 + 1.03e8T + 2.54e16T^{2} \) |
| 37 | \( 1 + 2.97e7T + 1.77e17T^{2} \) |
| 41 | \( 1 + 4.22e8T + 5.50e17T^{2} \) |
| 43 | \( 1 - 3.51e8T + 9.29e17T^{2} \) |
| 47 | \( 1 - 1.85e8T + 2.47e18T^{2} \) |
| 53 | \( 1 + 3.89e9T + 9.26e18T^{2} \) |
| 59 | \( 1 + 6.93e9T + 3.01e19T^{2} \) |
| 61 | \( 1 - 3.21e9T + 4.35e19T^{2} \) |
| 67 | \( 1 - 1.07e10T + 1.22e20T^{2} \) |
| 71 | \( 1 - 1.05e10T + 2.31e20T^{2} \) |
| 73 | \( 1 + 1.93e10T + 3.13e20T^{2} \) |
| 79 | \( 1 + 4.97e9T + 7.47e20T^{2} \) |
| 83 | \( 1 - 1.40e9T + 1.28e21T^{2} \) |
| 89 | \( 1 + 6.73e10T + 2.77e21T^{2} \) |
| 97 | \( 1 - 9.77e10T + 7.15e21T^{2} \) |
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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
−9.653787798888122593729770655500, −8.294455687016033380192838621953, −7.74110021813602793659729239657, −7.27564615053453142569207536763, −5.67575246196268203662442319102, −4.65239348388742775918605208329, −3.25015098914109558519793863635, −1.89071680983690592381830824648, −0.68957815567530466681059461554, 0,
0.68957815567530466681059461554, 1.89071680983690592381830824648, 3.25015098914109558519793863635, 4.65239348388742775918605208329, 5.67575246196268203662442319102, 7.27564615053453142569207536763, 7.74110021813602793659729239657, 8.294455687016033380192838621953, 9.653787798888122593729770655500
