| L(s) = 1 | − 5.41·2-s + 6.74·3-s − 2.69·4-s + 25·5-s − 36.5·6-s + 167.·7-s + 187.·8-s − 197.·9-s − 135.·10-s − 375.·11-s − 18.1·12-s + 353.·13-s − 908.·14-s + 168.·15-s − 930.·16-s + 1.42e3·17-s + 1.06e3·18-s − 1.48e3·19-s − 67.3·20-s + 1.13e3·21-s + 2.03e3·22-s − 529·23-s + 1.26e3·24-s + 625·25-s − 1.91e3·26-s − 2.97e3·27-s − 452.·28-s + ⋯ |
| L(s) = 1 | − 0.957·2-s + 0.432·3-s − 0.0841·4-s + 0.447·5-s − 0.414·6-s + 1.29·7-s + 1.03·8-s − 0.812·9-s − 0.427·10-s − 0.935·11-s − 0.0364·12-s + 0.580·13-s − 1.23·14-s + 0.193·15-s − 0.908·16-s + 1.19·17-s + 0.777·18-s − 0.942·19-s − 0.0376·20-s + 0.560·21-s + 0.894·22-s − 0.208·23-s + 0.449·24-s + 0.200·25-s − 0.555·26-s − 0.784·27-s − 0.108·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.377657942\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.377657942\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 25T \) |
| 23 | \( 1 + 529T \) |
| good | 2 | \( 1 + 5.41T + 32T^{2} \) |
| 3 | \( 1 - 6.74T + 243T^{2} \) |
| 7 | \( 1 - 167.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 375.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 353.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.42e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.48e3T + 2.47e6T^{2} \) |
| 29 | \( 1 - 7.66e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 1.83e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.26e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.30e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.97e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.00e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.11e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.11e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.49e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 6.83e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 6.98e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.12e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 6.88e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.80e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.55e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.60e4T + 8.58e9T^{2} \) |
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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
−12.67728382929458726831583603828, −11.16261441735733587116024991812, −10.44761894763017976676710726329, −9.255093954119117035938854492907, −8.218752593874531889942391240377, −7.84118524292513314595097230540, −5.81194460516726791463359815403, −4.52283091046563122034739736845, −2.47304259523790880671345865537, −0.978688998892845808788436173562,
0.978688998892845808788436173562, 2.47304259523790880671345865537, 4.52283091046563122034739736845, 5.81194460516726791463359815403, 7.84118524292513314595097230540, 8.218752593874531889942391240377, 9.255093954119117035938854492907, 10.44761894763017976676710726329, 11.16261441735733587116024991812, 12.67728382929458726831583603828
