L(s) = 1 | + 2-s + 0.630·3-s + 4-s + 3.47·5-s + 0.630·6-s − 7-s + 8-s − 2.60·9-s + 3.47·10-s − 2.00·11-s + 0.630·12-s − 1.94·13-s − 14-s + 2.19·15-s + 16-s + 3.57·17-s − 2.60·18-s + 7.18·19-s + 3.47·20-s − 0.630·21-s − 2.00·22-s − 3.84·23-s + 0.630·24-s + 7.07·25-s − 1.94·26-s − 3.53·27-s − 28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.364·3-s + 0.5·4-s + 1.55·5-s + 0.257·6-s − 0.377·7-s + 0.353·8-s − 0.867·9-s + 1.09·10-s − 0.603·11-s + 0.182·12-s − 0.539·13-s − 0.267·14-s + 0.565·15-s + 0.250·16-s + 0.865·17-s − 0.613·18-s + 1.64·19-s + 0.777·20-s − 0.137·21-s − 0.426·22-s − 0.801·23-s + 0.128·24-s + 1.41·25-s − 0.381·26-s − 0.679·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.602740257\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.602740257\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 431 | \( 1 + T \) |
good | 3 | \( 1 - 0.630T + 3T^{2} \) |
| 5 | \( 1 - 3.47T + 5T^{2} \) |
| 11 | \( 1 + 2.00T + 11T^{2} \) |
| 13 | \( 1 + 1.94T + 13T^{2} \) |
| 17 | \( 1 - 3.57T + 17T^{2} \) |
| 19 | \( 1 - 7.18T + 19T^{2} \) |
| 23 | \( 1 + 3.84T + 23T^{2} \) |
| 29 | \( 1 - 7.36T + 29T^{2} \) |
| 31 | \( 1 + 2.56T + 31T^{2} \) |
| 37 | \( 1 - 8.12T + 37T^{2} \) |
| 41 | \( 1 - 6.82T + 41T^{2} \) |
| 43 | \( 1 - 5.87T + 43T^{2} \) |
| 47 | \( 1 - 6.49T + 47T^{2} \) |
| 53 | \( 1 + 4.20T + 53T^{2} \) |
| 59 | \( 1 + 10.5T + 59T^{2} \) |
| 61 | \( 1 + 5.84T + 61T^{2} \) |
| 67 | \( 1 - 2.15T + 67T^{2} \) |
| 71 | \( 1 + 0.416T + 71T^{2} \) |
| 73 | \( 1 - 15.2T + 73T^{2} \) |
| 79 | \( 1 + 11.7T + 79T^{2} \) |
| 83 | \( 1 - 11.5T + 83T^{2} \) |
| 89 | \( 1 + 1.88T + 89T^{2} \) |
| 97 | \( 1 + 7.27T + 97T^{2} \) |
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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
−7.83671019652302985507352794195, −7.45655293729456986421791282508, −6.29264254971916714114853996649, −5.86550496697101641937520596936, −5.38436192690207679492764705782, −4.61451806188428044539812000356, −3.38239588930568265942520052387, −2.75122661452129597597845874730, −2.24949302748835041123076214188, −1.01713697055146651439918183417,
1.01713697055146651439918183417, 2.24949302748835041123076214188, 2.75122661452129597597845874730, 3.38239588930568265942520052387, 4.61451806188428044539812000356, 5.38436192690207679492764705782, 5.86550496697101641937520596936, 6.29264254971916714114853996649, 7.45655293729456986421791282508, 7.83671019652302985507352794195