| L(s) = 1 | − 0.902·2-s − 3.32·3-s − 1.18·4-s − 0.938·5-s + 3.00·6-s + 2.87·8-s + 8.05·9-s + 0.846·10-s + 5.46·11-s + 3.94·12-s − 2.52·13-s + 3.12·15-s − 0.223·16-s + 6.71·17-s − 7.27·18-s + 0.857·19-s + 1.11·20-s − 4.92·22-s + 4.76·23-s − 9.55·24-s − 4.11·25-s + 2.27·26-s − 16.8·27-s + 3.28·29-s − 2.81·30-s + 0.119·31-s − 5.54·32-s + ⋯ |
| L(s) = 1 | − 0.638·2-s − 1.91·3-s − 0.592·4-s − 0.419·5-s + 1.22·6-s + 1.01·8-s + 2.68·9-s + 0.267·10-s + 1.64·11-s + 1.13·12-s − 0.700·13-s + 0.805·15-s − 0.0558·16-s + 1.62·17-s − 1.71·18-s + 0.196·19-s + 0.248·20-s − 1.05·22-s + 0.993·23-s − 1.95·24-s − 0.823·25-s + 0.446·26-s − 3.23·27-s + 0.610·29-s − 0.514·30-s + 0.0214·31-s − 0.980·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2009 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5450765087\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5450765087\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 2 | \( 1 + 0.902T + 2T^{2} \) |
| 3 | \( 1 + 3.32T + 3T^{2} \) |
| 5 | \( 1 + 0.938T + 5T^{2} \) |
| 11 | \( 1 - 5.46T + 11T^{2} \) |
| 13 | \( 1 + 2.52T + 13T^{2} \) |
| 17 | \( 1 - 6.71T + 17T^{2} \) |
| 19 | \( 1 - 0.857T + 19T^{2} \) |
| 23 | \( 1 - 4.76T + 23T^{2} \) |
| 29 | \( 1 - 3.28T + 29T^{2} \) |
| 31 | \( 1 - 0.119T + 31T^{2} \) |
| 37 | \( 1 + 3.81T + 37T^{2} \) |
| 43 | \( 1 - 1.64T + 43T^{2} \) |
| 47 | \( 1 - 4.73T + 47T^{2} \) |
| 53 | \( 1 - 2.24T + 53T^{2} \) |
| 59 | \( 1 - 4.44T + 59T^{2} \) |
| 61 | \( 1 - 11.9T + 61T^{2} \) |
| 67 | \( 1 + 11.7T + 67T^{2} \) |
| 71 | \( 1 - 9.78T + 71T^{2} \) |
| 73 | \( 1 + 3.71T + 73T^{2} \) |
| 79 | \( 1 + 2.81T + 79T^{2} \) |
| 83 | \( 1 - 3.41T + 83T^{2} \) |
| 89 | \( 1 + 14.2T + 89T^{2} \) |
| 97 | \( 1 + 8.40T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.511077277155636817244370887262, −8.388186443370357334944605929513, −7.33570829257955162402243602341, −6.96803767452954543951308585239, −5.87566907885387357277738690712, −5.22723265103374760467904138147, −4.39953407093382832577656093746, −3.70877989393177989507398441457, −1.40715412428196910238800925027, −0.69288411430431237962646359197,
0.69288411430431237962646359197, 1.40715412428196910238800925027, 3.70877989393177989507398441457, 4.39953407093382832577656093746, 5.22723265103374760467904138147, 5.87566907885387357277738690712, 6.96803767452954543951308585239, 7.33570829257955162402243602341, 8.388186443370357334944605929513, 9.511077277155636817244370887262
