| L(s) = 1 | − 2.73·2-s + 1.50·3-s + 5.45·4-s − 3.60·5-s − 4.10·6-s + 2.19·7-s − 9.43·8-s − 0.737·9-s + 9.84·10-s + 3.45·11-s + 8.20·12-s + 13-s − 5.98·14-s − 5.42·15-s + 14.8·16-s + 0.177·17-s + 2.01·18-s + 2.34·19-s − 19.6·20-s + 3.29·21-s − 9.44·22-s − 6.25·23-s − 14.1·24-s + 8.00·25-s − 2.73·26-s − 5.62·27-s + 11.9·28-s + ⋯ |
| L(s) = 1 | − 1.93·2-s + 0.868·3-s + 2.72·4-s − 1.61·5-s − 1.67·6-s + 0.828·7-s − 3.33·8-s − 0.245·9-s + 3.11·10-s + 1.04·11-s + 2.36·12-s + 0.277·13-s − 1.59·14-s − 1.40·15-s + 3.71·16-s + 0.0430·17-s + 0.474·18-s + 0.537·19-s − 4.39·20-s + 0.719·21-s − 2.01·22-s − 1.30·23-s − 2.89·24-s + 1.60·25-s − 0.535·26-s − 1.08·27-s + 2.26·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8047 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8047 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 - T \) |
| 619 | \( 1 - T \) |
| good | 2 | \( 1 + 2.73T + 2T^{2} \) |
| 3 | \( 1 - 1.50T + 3T^{2} \) |
| 5 | \( 1 + 3.60T + 5T^{2} \) |
| 7 | \( 1 - 2.19T + 7T^{2} \) |
| 11 | \( 1 - 3.45T + 11T^{2} \) |
| 17 | \( 1 - 0.177T + 17T^{2} \) |
| 19 | \( 1 - 2.34T + 19T^{2} \) |
| 23 | \( 1 + 6.25T + 23T^{2} \) |
| 29 | \( 1 + 6.36T + 29T^{2} \) |
| 31 | \( 1 + 0.718T + 31T^{2} \) |
| 37 | \( 1 - 7.99T + 37T^{2} \) |
| 41 | \( 1 - 9.73T + 41T^{2} \) |
| 43 | \( 1 - 4.13T + 43T^{2} \) |
| 47 | \( 1 + 2.61T + 47T^{2} \) |
| 53 | \( 1 + 9.07T + 53T^{2} \) |
| 59 | \( 1 - 4.03T + 59T^{2} \) |
| 61 | \( 1 + 0.809T + 61T^{2} \) |
| 67 | \( 1 + 13.0T + 67T^{2} \) |
| 71 | \( 1 - 10.3T + 71T^{2} \) |
| 73 | \( 1 + 14.3T + 73T^{2} \) |
| 79 | \( 1 + 6.47T + 79T^{2} \) |
| 83 | \( 1 - 5.29T + 83T^{2} \) |
| 89 | \( 1 - 8.97T + 89T^{2} \) |
| 97 | \( 1 + 5.00T + 97T^{2} \) |
| 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
−7.76768834233504241453079238424, −7.48797746550365996515155985098, −6.46533636646376643832522364622, −5.75554912880446836845586681673, −4.30183818887180853215898456257, −3.65428406072062455164812748327, −2.88624351877326581004465517029, −1.94705491456334666542461836145, −1.10129529354204782222211950231, 0,
1.10129529354204782222211950231, 1.94705491456334666542461836145, 2.88624351877326581004465517029, 3.65428406072062455164812748327, 4.30183818887180853215898456257, 5.75554912880446836845586681673, 6.46533636646376643832522364622, 7.48797746550365996515155985098, 7.76768834233504241453079238424
