| L(s) = 1 | − 4·2-s − 9·3-s + 16·4-s + 17.9·5-s + 36·6-s + 177.·7-s − 64·8-s + 81·9-s − 71.7·10-s + 129.·11-s − 144·12-s − 1.17e3·13-s − 708.·14-s − 161.·15-s + 256·16-s + 756.·17-s − 324·18-s − 227.·19-s + 286.·20-s − 1.59e3·21-s − 516.·22-s − 3.36e3·23-s + 576·24-s − 2.80e3·25-s + 4.68e3·26-s − 729·27-s + 2.83e3·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.320·5-s + 0.408·6-s + 1.36·7-s − 0.353·8-s + 0.333·9-s − 0.226·10-s + 0.321·11-s − 0.288·12-s − 1.92·13-s − 0.965·14-s − 0.185·15-s + 0.250·16-s + 0.634·17-s − 0.235·18-s − 0.144·19-s + 0.160·20-s − 0.788·21-s − 0.227·22-s − 1.32·23-s + 0.204·24-s − 0.897·25-s + 1.35·26-s − 0.192·27-s + 0.682·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 4T \) |
| 3 | \( 1 + 9T \) |
| 59 | \( 1 + 3.48e3T \) |
| good | 5 | \( 1 - 17.9T + 3.12e3T^{2} \) |
| 7 | \( 1 - 177.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 129.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 1.17e3T + 3.71e5T^{2} \) |
| 17 | \( 1 - 756.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 227.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.36e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 449.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 4.61e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.48e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 6.23e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.39e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.19e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.53e4T + 4.18e8T^{2} \) |
| 61 | \( 1 + 4.60e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.74e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 4.09e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.02e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 1.08e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 6.91e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.04e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.41e5T + 8.58e9T^{2} \) |
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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
−10.02081355892882734449219771995, −9.549012037314903950035500066498, −8.051646768044314366407721632427, −7.62845752275272186088311475445, −6.35780587614735535368633582625, −5.29932535983936717596813679597, −4.36078669500929692297891483747, −2.40709138550975509227927535050, −1.41427673645440001448025635287, 0,
1.41427673645440001448025635287, 2.40709138550975509227927535050, 4.36078669500929692297891483747, 5.29932535983936717596813679597, 6.35780587614735535368633582625, 7.62845752275272186088311475445, 8.051646768044314366407721632427, 9.549012037314903950035500066498, 10.02081355892882734449219771995
