| L(s) = 1 | + 2-s − 0.440·3-s + 4-s − 5-s − 0.440·6-s + 3.12·7-s + 8-s − 2.80·9-s − 10-s − 11-s − 0.440·12-s + 5.02·13-s + 3.12·14-s + 0.440·15-s + 16-s − 0.113·17-s − 2.80·18-s − 6.57·19-s − 20-s − 1.37·21-s − 22-s + 2.34·23-s − 0.440·24-s + 25-s + 5.02·26-s + 2.55·27-s + 3.12·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.254·3-s + 0.5·4-s − 0.447·5-s − 0.179·6-s + 1.18·7-s + 0.353·8-s − 0.935·9-s − 0.316·10-s − 0.301·11-s − 0.127·12-s + 1.39·13-s + 0.835·14-s + 0.113·15-s + 0.250·16-s − 0.0275·17-s − 0.661·18-s − 1.50·19-s − 0.223·20-s − 0.300·21-s − 0.213·22-s + 0.489·23-s − 0.0899·24-s + 0.200·25-s + 0.985·26-s + 0.492·27-s + 0.591·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 73 | \( 1 - T \) |
| good | 3 | \( 1 + 0.440T + 3T^{2} \) |
| 7 | \( 1 - 3.12T + 7T^{2} \) |
| 13 | \( 1 - 5.02T + 13T^{2} \) |
| 17 | \( 1 + 0.113T + 17T^{2} \) |
| 19 | \( 1 + 6.57T + 19T^{2} \) |
| 23 | \( 1 - 2.34T + 23T^{2} \) |
| 29 | \( 1 + 10.4T + 29T^{2} \) |
| 31 | \( 1 + 1.18T + 31T^{2} \) |
| 37 | \( 1 + 9.28T + 37T^{2} \) |
| 41 | \( 1 - 8.98T + 41T^{2} \) |
| 43 | \( 1 + 12.0T + 43T^{2} \) |
| 47 | \( 1 + 10.6T + 47T^{2} \) |
| 53 | \( 1 + 4.37T + 53T^{2} \) |
| 59 | \( 1 - 4.55T + 59T^{2} \) |
| 61 | \( 1 + 13.4T + 61T^{2} \) |
| 67 | \( 1 - 11.8T + 67T^{2} \) |
| 71 | \( 1 - 6.49T + 71T^{2} \) |
| 79 | \( 1 + 1.20T + 79T^{2} \) |
| 83 | \( 1 - 0.720T + 83T^{2} \) |
| 89 | \( 1 - 17.7T + 89T^{2} \) |
| 97 | \( 1 - 3.30T + 97T^{2} \) |
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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
−7.51507139542696512774889783956, −6.57178994811792411934647001482, −6.07190217917792202516594545625, −5.21601092745895535448316903093, −4.84741565088892476156756538354, −3.83913668055566579983034706315, −3.38039305819962267190021758116, −2.20810307883413700486570458127, −1.48077790709630849671253831866, 0,
1.48077790709630849671253831866, 2.20810307883413700486570458127, 3.38039305819962267190021758116, 3.83913668055566579983034706315, 4.84741565088892476156756538354, 5.21601092745895535448316903093, 6.07190217917792202516594545625, 6.57178994811792411934647001482, 7.51507139542696512774889783956
