| L(s) = 1 | − 2.65·2-s + 1.08·3-s + 5.03·4-s + 0.484·5-s − 2.87·6-s + 1.56·7-s − 8.04·8-s − 1.82·9-s − 1.28·10-s + 11-s + 5.45·12-s + 1.02·13-s − 4.15·14-s + 0.524·15-s + 11.2·16-s + 17-s + 4.84·18-s − 7.33·19-s + 2.43·20-s + 1.69·21-s − 2.65·22-s + 4.79·23-s − 8.70·24-s − 4.76·25-s − 2.70·26-s − 5.22·27-s + 7.89·28-s + ⋯ |
| L(s) = 1 | − 1.87·2-s + 0.625·3-s + 2.51·4-s + 0.216·5-s − 1.17·6-s + 0.592·7-s − 2.84·8-s − 0.609·9-s − 0.406·10-s + 0.301·11-s + 1.57·12-s + 0.282·13-s − 1.11·14-s + 0.135·15-s + 2.81·16-s + 0.242·17-s + 1.14·18-s − 1.68·19-s + 0.545·20-s + 0.370·21-s − 0.565·22-s + 0.999·23-s − 1.77·24-s − 0.953·25-s − 0.530·26-s − 1.00·27-s + 1.49·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8858502982\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8858502982\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 43 | \( 1 - T \) |
| good | 2 | \( 1 + 2.65T + 2T^{2} \) |
| 3 | \( 1 - 1.08T + 3T^{2} \) |
| 5 | \( 1 - 0.484T + 5T^{2} \) |
| 7 | \( 1 - 1.56T + 7T^{2} \) |
| 13 | \( 1 - 1.02T + 13T^{2} \) |
| 19 | \( 1 + 7.33T + 19T^{2} \) |
| 23 | \( 1 - 4.79T + 23T^{2} \) |
| 29 | \( 1 + 6.14T + 29T^{2} \) |
| 31 | \( 1 + 8.14T + 31T^{2} \) |
| 37 | \( 1 - 7.46T + 37T^{2} \) |
| 41 | \( 1 + 10.1T + 41T^{2} \) |
| 47 | \( 1 - 1.41T + 47T^{2} \) |
| 53 | \( 1 - 10.8T + 53T^{2} \) |
| 59 | \( 1 + 1.27T + 59T^{2} \) |
| 61 | \( 1 - 5.68T + 61T^{2} \) |
| 67 | \( 1 - 11.3T + 67T^{2} \) |
| 71 | \( 1 - 4.89T + 71T^{2} \) |
| 73 | \( 1 - 1.41T + 73T^{2} \) |
| 79 | \( 1 + 8.10T + 79T^{2} \) |
| 83 | \( 1 - 6.65T + 83T^{2} \) |
| 89 | \( 1 - 11.4T + 89T^{2} \) |
| 97 | \( 1 - 1.82T + 97T^{2} \) |
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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
−8.131904194881453308770305437392, −7.43524870628579177459104031839, −6.77356916787611066249355029350, −6.02232317252311916148645335987, −5.34177696744302182296243345963, −3.98738552522290996805740826764, −3.18077297204202379635218539951, −2.06937371882322219043821645779, −1.90401146346328139858153796640, −0.57846958815398078868967536694,
0.57846958815398078868967536694, 1.90401146346328139858153796640, 2.06937371882322219043821645779, 3.18077297204202379635218539951, 3.98738552522290996805740826764, 5.34177696744302182296243345963, 6.02232317252311916148645335987, 6.77356916787611066249355029350, 7.43524870628579177459104031839, 8.131904194881453308770305437392
