| L(s) = 1 | + 1.31·2-s − 1.92·3-s − 0.270·4-s + 5-s − 2.53·6-s + 2.58·7-s − 2.98·8-s + 0.702·9-s + 1.31·10-s − 11-s + 0.520·12-s − 5.20·13-s + 3.39·14-s − 1.92·15-s − 3.38·16-s − 5.07·17-s + 0.923·18-s − 1.31·19-s − 0.270·20-s − 4.96·21-s − 1.31·22-s + 6.09·23-s + 5.74·24-s + 25-s − 6.83·26-s + 4.42·27-s − 0.698·28-s + ⋯ |
| L(s) = 1 | + 0.929·2-s − 1.11·3-s − 0.135·4-s + 0.447·5-s − 1.03·6-s + 0.975·7-s − 1.05·8-s + 0.234·9-s + 0.415·10-s − 0.301·11-s + 0.150·12-s − 1.44·13-s + 0.906·14-s − 0.496·15-s − 0.846·16-s − 1.22·17-s + 0.217·18-s − 0.302·19-s − 0.0605·20-s − 1.08·21-s − 0.280·22-s + 1.27·23-s + 1.17·24-s + 0.200·25-s − 1.34·26-s + 0.850·27-s − 0.131·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.455563782\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.455563782\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 73 | \( 1 + T \) |
| good | 2 | \( 1 - 1.31T + 2T^{2} \) |
| 3 | \( 1 + 1.92T + 3T^{2} \) |
| 7 | \( 1 - 2.58T + 7T^{2} \) |
| 13 | \( 1 + 5.20T + 13T^{2} \) |
| 17 | \( 1 + 5.07T + 17T^{2} \) |
| 19 | \( 1 + 1.31T + 19T^{2} \) |
| 23 | \( 1 - 6.09T + 23T^{2} \) |
| 29 | \( 1 - 7.71T + 29T^{2} \) |
| 31 | \( 1 + 4.14T + 31T^{2} \) |
| 37 | \( 1 + 1.83T + 37T^{2} \) |
| 41 | \( 1 + 0.766T + 41T^{2} \) |
| 43 | \( 1 - 5.47T + 43T^{2} \) |
| 47 | \( 1 - 11.6T + 47T^{2} \) |
| 53 | \( 1 + 8.14T + 53T^{2} \) |
| 59 | \( 1 - 3.47T + 59T^{2} \) |
| 61 | \( 1 - 8.11T + 61T^{2} \) |
| 67 | \( 1 + 3.31T + 67T^{2} \) |
| 71 | \( 1 + 12.2T + 71T^{2} \) |
| 79 | \( 1 - 11.9T + 79T^{2} \) |
| 83 | \( 1 + 3.67T + 83T^{2} \) |
| 89 | \( 1 - 4.35T + 89T^{2} \) |
| 97 | \( 1 - 13.7T + 97T^{2} \) |
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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
−8.605222393233465981916905805668, −7.45101153349344453001352079612, −6.72203157182984597883486370200, −5.99230836043210021428021077464, −5.22983026927507305258638486452, −4.83074905111074478774559075732, −4.37113523701258138030989827206, −2.95278607920715708348157707640, −2.16854168061569148232031190063, −0.61608908832641065045674379031,
0.61608908832641065045674379031, 2.16854168061569148232031190063, 2.95278607920715708348157707640, 4.37113523701258138030989827206, 4.83074905111074478774559075732, 5.22983026927507305258638486452, 5.99230836043210021428021077464, 6.72203157182984597883486370200, 7.45101153349344453001352079612, 8.605222393233465981916905805668
