| L(s) = 1 | + 2.38·2-s + 3.66·4-s + 5-s + 2.24·7-s + 3.97·8-s + 2.38·10-s + 5.33·14-s + 2.12·16-s + 0.509·17-s + 2.24·19-s + 3.66·20-s + 4.45·23-s + 25-s + 8.22·28-s − 3.46·29-s − 7.24·31-s − 2.89·32-s + 1.21·34-s + 2.24·35-s + 10.7·37-s + 5.33·38-s + 3.97·40-s + 7.94·41-s + 6.05·43-s + 10.6·46-s + 12.2·47-s − 1.97·49-s + ⋯ |
| L(s) = 1 | + 1.68·2-s + 1.83·4-s + 0.447·5-s + 0.847·7-s + 1.40·8-s + 0.752·10-s + 1.42·14-s + 0.530·16-s + 0.123·17-s + 0.514·19-s + 0.820·20-s + 0.928·23-s + 0.200·25-s + 1.55·28-s − 0.643·29-s − 1.30·31-s − 0.511·32-s + 0.208·34-s + 0.378·35-s + 1.77·37-s + 0.865·38-s + 0.628·40-s + 1.24·41-s + 0.924·43-s + 1.56·46-s + 1.78·47-s − 0.281·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.038923970\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.038923970\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 2.38T + 2T^{2} \) |
| 7 | \( 1 - 2.24T + 7T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 0.509T + 17T^{2} \) |
| 19 | \( 1 - 2.24T + 19T^{2} \) |
| 23 | \( 1 - 4.45T + 23T^{2} \) |
| 29 | \( 1 + 3.46T + 29T^{2} \) |
| 31 | \( 1 + 7.24T + 31T^{2} \) |
| 37 | \( 1 - 10.7T + 37T^{2} \) |
| 41 | \( 1 - 7.94T + 41T^{2} \) |
| 43 | \( 1 - 6.05T + 43T^{2} \) |
| 47 | \( 1 - 12.2T + 47T^{2} \) |
| 53 | \( 1 + 3.79T + 53T^{2} \) |
| 59 | \( 1 + 4.67T + 59T^{2} \) |
| 61 | \( 1 - 7.79T + 61T^{2} \) |
| 67 | \( 1 + 8.79T + 67T^{2} \) |
| 71 | \( 1 - 16.6T + 71T^{2} \) |
| 73 | \( 1 + 12.7T + 73T^{2} \) |
| 79 | \( 1 - 6.49T + 79T^{2} \) |
| 83 | \( 1 + 12.9T + 83T^{2} \) |
| 89 | \( 1 - 0.180T + 89T^{2} \) |
| 97 | \( 1 + 10.6T + 97T^{2} \) |
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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
−7.74899507668353545129820310751, −7.36908135583860990740659797103, −6.46195431943793771862783575614, −5.67501463205552484392428322844, −5.34228806104660944473393694793, −4.48672456022439222054131163585, −3.92341489969317228225152237213, −2.91827259762227594056278687492, −2.25413257360884070837880843258, −1.21299121954803015005032868484,
1.21299121954803015005032868484, 2.25413257360884070837880843258, 2.91827259762227594056278687492, 3.92341489969317228225152237213, 4.48672456022439222054131163585, 5.34228806104660944473393694793, 5.67501463205552484392428322844, 6.46195431943793771862783575614, 7.36908135583860990740659797103, 7.74899507668353545129820310751
