| L(s) = 1 | + 117·2-s + 9.59e3·4-s + 1.56e4·5-s + 8.72e4·7-s + 6.43e5·8-s + 5.31e5·9-s + 1.82e6·10-s + 1.77e6·11-s − 9.62e6·13-s + 1.02e7·14-s + 3.59e7·16-s − 3.57e7·17-s + 6.21e7·18-s + 1.49e8·20-s + 2.07e8·22-s + 2.44e8·25-s − 1.12e9·26-s + 8.37e8·28-s + 3.46e8·31-s + 1.57e9·32-s − 4.18e9·34-s + 1.36e9·35-s + 5.09e9·36-s + 1.00e10·40-s + 7.56e9·43-s + 1.69e10·44-s + 8.30e9·45-s + ⋯ |
| L(s) = 1 | + 1.82·2-s + 2.34·4-s + 5-s + 0.741·7-s + 2.45·8-s + 9-s + 1.82·10-s + 11-s − 1.99·13-s + 1.35·14-s + 2.14·16-s − 1.48·17-s + 1.82·18-s + 2.34·20-s + 1.82·22-s + 25-s − 3.64·26-s + 1.73·28-s + 0.390·31-s + 1.46·32-s − 2.70·34-s + 0.741·35-s + 2.34·36-s + 2.45·40-s + 1.19·43-s + 2.34·44-s + 45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{13}{2})\) |
\(\approx\) |
\(8.932177383\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.932177383\) |
| \(L(7)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - p^{6} T \) |
| 11 | \( 1 - p^{6} T \) |
| good | 2 | \( 1 - 117 T + p^{12} T^{2} \) |
| 3 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 7 | \( 1 - 87282 T + p^{12} T^{2} \) |
| 13 | \( 1 + 9629118 T + p^{12} T^{2} \) |
| 17 | \( 1 + 35761518 T + p^{12} T^{2} \) |
| 19 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 23 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 29 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 31 | \( 1 - 346700482 T + p^{12} T^{2} \) |
| 37 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 41 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 43 | \( 1 - 7565541282 T + p^{12} T^{2} \) |
| 47 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 53 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 59 | \( 1 + 84345242798 T + p^{12} T^{2} \) |
| 61 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 67 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 71 | \( 1 - 240883653922 T + p^{12} T^{2} \) |
| 73 | \( 1 + 297569550798 T + p^{12} T^{2} \) |
| 79 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
| 83 | \( 1 - 579981676482 T + p^{12} T^{2} \) |
| 89 | \( 1 + 991704621598 T + p^{12} T^{2} \) |
| 97 | \( ( 1 - p^{6} T )( 1 + p^{6} T ) \) |
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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
−12.85710792878148905544625607651, −12.04167073937085698493251992332, −10.81824301937244569001785772743, −9.488725778617777594434665097215, −7.27024974894912887227757229945, −6.38732269612131665315951293117, −4.96605840675187748484909205564, −4.33726001434113555830997636613, −2.51104141360592383742186806575, −1.64436595698531083422304295128,
1.64436595698531083422304295128, 2.51104141360592383742186806575, 4.33726001434113555830997636613, 4.96605840675187748484909205564, 6.38732269612131665315951293117, 7.27024974894912887227757229945, 9.488725778617777594434665097215, 10.81824301937244569001785772743, 12.04167073937085698493251992332, 12.85710792878148905544625607651
