| L(s) = 1 | − 9.47·2-s + 20.3·3-s + 57.7·4-s + 25·5-s − 192.·6-s − 244.·8-s + 169.·9-s − 236.·10-s − 630.·11-s + 1.17e3·12-s − 192.·13-s + 507.·15-s + 464.·16-s − 847.·17-s − 1.60e3·18-s + 3.12e3·19-s + 1.44e3·20-s + 5.97e3·22-s + 1.80e3·23-s − 4.95e3·24-s + 625·25-s + 1.82e3·26-s − 1.50e3·27-s + 8.32e3·29-s − 4.80e3·30-s + 3.41e3·31-s + 3.41e3·32-s + ⋯ |
| L(s) = 1 | − 1.67·2-s + 1.30·3-s + 1.80·4-s + 0.447·5-s − 2.18·6-s − 1.34·8-s + 0.695·9-s − 0.749·10-s − 1.57·11-s + 2.35·12-s − 0.315·13-s + 0.582·15-s + 0.453·16-s − 0.711·17-s − 1.16·18-s + 1.98·19-s + 0.807·20-s + 2.63·22-s + 0.711·23-s − 1.75·24-s + 0.200·25-s + 0.529·26-s − 0.396·27-s + 1.83·29-s − 0.975·30-s + 0.638·31-s + 0.589·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.419443400\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.419443400\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 25T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 9.47T + 32T^{2} \) |
| 3 | \( 1 - 20.3T + 243T^{2} \) |
| 11 | \( 1 + 630.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 192.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 847.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 3.12e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.80e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 8.32e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 3.41e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 9.00e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.28e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 5.80e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 9.62e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.88e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.56e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.11e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.49e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.37e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.84e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 9.69e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.04e5T + 3.93e9T^{2} \) |
| 89 | \( 1 + 4.68e3T + 5.58e9T^{2} \) |
| 97 | \( 1 + 4.35e4T + 8.58e9T^{2} \) |
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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
−10.70278253950659436575106897426, −9.918635551299558394835829314185, −9.279047139535522606176959844710, −8.331043778434513021382319436275, −7.78676288834412124143941784717, −6.81273364605149797857257521431, −5.10752035604625008472659818194, −2.95237342741049068054141698051, −2.31707883362312597582521424510, −0.842855322886977833845563700861,
0.842855322886977833845563700861, 2.31707883362312597582521424510, 2.95237342741049068054141698051, 5.10752035604625008472659818194, 6.81273364605149797857257521431, 7.78676288834412124143941784717, 8.331043778434513021382319436275, 9.279047139535522606176959844710, 9.918635551299558394835829314185, 10.70278253950659436575106897426
