| L(s) = 1 | − 13.6·3-s − 91.2·5-s − 63.7·7-s − 57.2·9-s − 686.·11-s − 962.·13-s + 1.24e3·15-s + 1.47e3·17-s − 1.32e3·19-s + 869.·21-s + 2.40e3·23-s + 5.20e3·25-s + 4.09e3·27-s − 1.73e3·29-s + 4.83e3·31-s + 9.35e3·33-s + 5.82e3·35-s + 9.14e3·37-s + 1.31e4·39-s − 6.86e3·41-s − 6.69e3·43-s + 5.22e3·45-s − 5.97e3·47-s − 1.27e4·49-s − 2.01e4·51-s + 6.46e3·53-s + 6.26e4·55-s + ⋯ |
| L(s) = 1 | − 0.874·3-s − 1.63·5-s − 0.491·7-s − 0.235·9-s − 1.71·11-s − 1.58·13-s + 1.42·15-s + 1.23·17-s − 0.843·19-s + 0.430·21-s + 0.946·23-s + 1.66·25-s + 1.08·27-s − 0.383·29-s + 0.904·31-s + 1.49·33-s + 0.803·35-s + 1.09·37-s + 1.38·39-s − 0.637·41-s − 0.551·43-s + 0.384·45-s − 0.394·47-s − 0.758·49-s − 1.08·51-s + 0.316·53-s + 2.79·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1028 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 257 | \( 1 - 6.60e4T \) |
| good | 3 | \( 1 + 13.6T + 243T^{2} \) |
| 5 | \( 1 + 91.2T + 3.12e3T^{2} \) |
| 7 | \( 1 + 63.7T + 1.68e4T^{2} \) |
| 11 | \( 1 + 686.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 962.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.47e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.32e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.40e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 1.73e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 4.83e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 9.14e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 6.86e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 6.69e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 5.97e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 6.46e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.22e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 5.53e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.72e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.84e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.39e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 6.95e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 9.54e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 4.44e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 6.75e4T + 8.58e9T^{2} \) |
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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
−8.560002634872575493576411565005, −7.78653585173712549676889271963, −7.30989254863583254166271264418, −6.23256526684603842174946123289, −5.09454321483721176612944029463, −4.72293490600202811508125999781, −3.33055953364979653120717360037, −2.63250798133191470857748062274, −0.60160332334945544776814723959, 0,
0.60160332334945544776814723959, 2.63250798133191470857748062274, 3.33055953364979653120717360037, 4.72293490600202811508125999781, 5.09454321483721176612944029463, 6.23256526684603842174946123289, 7.30989254863583254166271264418, 7.78653585173712549676889271963, 8.560002634872575493576411565005
