| L(s) = 1 | + 2-s − 2.19·3-s + 4-s + 1.33·5-s − 2.19·6-s − 4.99·7-s + 8-s + 1.80·9-s + 1.33·10-s + 3.23·11-s − 2.19·12-s + 3.93·13-s − 4.99·14-s − 2.92·15-s + 16-s + 2.30·17-s + 1.80·18-s − 19-s + 1.33·20-s + 10.9·21-s + 3.23·22-s + 2.41·23-s − 2.19·24-s − 3.21·25-s + 3.93·26-s + 2.61·27-s − 4.99·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.26·3-s + 0.5·4-s + 0.597·5-s − 0.895·6-s − 1.88·7-s + 0.353·8-s + 0.602·9-s + 0.422·10-s + 0.975·11-s − 0.632·12-s + 1.09·13-s − 1.33·14-s − 0.756·15-s + 0.250·16-s + 0.558·17-s + 0.425·18-s − 0.229·19-s + 0.298·20-s + 2.38·21-s + 0.689·22-s + 0.502·23-s − 0.447·24-s − 0.642·25-s + 0.771·26-s + 0.503·27-s − 0.943·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.812656462\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.812656462\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| 211 | \( 1 + T \) |
| good | 3 | \( 1 + 2.19T + 3T^{2} \) |
| 5 | \( 1 - 1.33T + 5T^{2} \) |
| 7 | \( 1 + 4.99T + 7T^{2} \) |
| 11 | \( 1 - 3.23T + 11T^{2} \) |
| 13 | \( 1 - 3.93T + 13T^{2} \) |
| 17 | \( 1 - 2.30T + 17T^{2} \) |
| 23 | \( 1 - 2.41T + 23T^{2} \) |
| 29 | \( 1 - 7.68T + 29T^{2} \) |
| 31 | \( 1 + 10.1T + 31T^{2} \) |
| 37 | \( 1 - 3.67T + 37T^{2} \) |
| 41 | \( 1 + 8.72T + 41T^{2} \) |
| 43 | \( 1 - 9.15T + 43T^{2} \) |
| 47 | \( 1 + 9.14T + 47T^{2} \) |
| 53 | \( 1 - 5.22T + 53T^{2} \) |
| 59 | \( 1 + 9.07T + 59T^{2} \) |
| 61 | \( 1 + 1.01T + 61T^{2} \) |
| 67 | \( 1 + 14.2T + 67T^{2} \) |
| 71 | \( 1 + 5.03T + 71T^{2} \) |
| 73 | \( 1 - 0.0673T + 73T^{2} \) |
| 79 | \( 1 - 17.2T + 79T^{2} \) |
| 83 | \( 1 + 7.67T + 83T^{2} \) |
| 89 | \( 1 - 16.3T + 89T^{2} \) |
| 97 | \( 1 + 11.3T + 97T^{2} \) |
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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
−7.40508793697920087687709248885, −6.69938228394360345283333986676, −6.12274884212901313523283617236, −6.06945275064089812303289136464, −5.26765096268114038069325078021, −4.30734974150123213386978938900, −3.52835124052068841632880658088, −2.98639311447330555940431298118, −1.67021689388451605050734148316, −0.64892563264492005995570478631,
0.64892563264492005995570478631, 1.67021689388451605050734148316, 2.98639311447330555940431298118, 3.52835124052068841632880658088, 4.30734974150123213386978938900, 5.26765096268114038069325078021, 6.06945275064089812303289136464, 6.12274884212901313523283617236, 6.69938228394360345283333986676, 7.40508793697920087687709248885
