| L(s) = 1 | + 3.11·2-s + 20.6·3-s − 22.3·4-s − 25·5-s + 64.1·6-s − 169.·8-s + 181.·9-s − 77.8·10-s − 142.·11-s − 459.·12-s + 634.·13-s − 515.·15-s + 187.·16-s + 1.30e3·17-s + 565.·18-s + 2.89e3·19-s + 557.·20-s − 442.·22-s + 280.·23-s − 3.48e3·24-s + 625·25-s + 1.97e3·26-s − 1.26e3·27-s + 4.41e3·29-s − 1.60e3·30-s + 2.88e3·31-s + 5.99e3·32-s + ⋯ |
| L(s) = 1 | + 0.550·2-s + 1.32·3-s − 0.697·4-s − 0.447·5-s + 0.727·6-s − 0.934·8-s + 0.747·9-s − 0.246·10-s − 0.353·11-s − 0.921·12-s + 1.04·13-s − 0.591·15-s + 0.182·16-s + 1.09·17-s + 0.411·18-s + 1.83·19-s + 0.311·20-s − 0.194·22-s + 0.110·23-s − 1.23·24-s + 0.200·25-s + 0.573·26-s − 0.333·27-s + 0.975·29-s − 0.325·30-s + 0.539·31-s + 1.03·32-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
| \(L(3)\) |
\(\approx\) |
\(3.424420949\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.424420949\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 - 3.11T + 32T^{2} \) |
| 3 | \( 1 - 20.6T + 243T^{2} \) |
| 11 | \( 1 + 142.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 634.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.30e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.89e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 280.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 4.41e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.88e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 6.03e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.94e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.11e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.35e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.60e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 4.61e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.30e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.39e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.25e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 8.15e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 4.22e3T + 3.07e9T^{2} \) |
| 83 | \( 1 - 6.61e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 5.83e3T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.56e5T + 8.58e9T^{2} \) |
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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
−11.42596602579180516879997677338, −9.955469345073947177898231313125, −9.200804903493673498278679632585, −8.251648162808363014139160103908, −7.65420381273231930618711787130, −5.97025556506507911029858277920, −4.76513208583739072536089881721, −3.51806482888419343949205601495, −3.00427803570023987068097500536, −1.00641748025717431402131588676,
1.00641748025717431402131588676, 3.00427803570023987068097500536, 3.51806482888419343949205601495, 4.76513208583739072536089881721, 5.97025556506507911029858277920, 7.65420381273231930618711787130, 8.251648162808363014139160103908, 9.200804903493673498278679632585, 9.955469345073947177898231313125, 11.42596602579180516879997677338
