| L(s) = 1 | + 0.476·2-s − 1.77·4-s + 3.87·5-s + 7-s − 1.79·8-s + 1.84·10-s − 5.00·11-s − 0.407·13-s + 0.476·14-s + 2.68·16-s − 0.710·17-s + 8.42·19-s − 6.87·20-s − 2.38·22-s + 5.39·23-s + 10.0·25-s − 0.194·26-s − 1.77·28-s − 8.15·29-s − 0.794·31-s + 4.87·32-s − 0.338·34-s + 3.87·35-s + 6.57·37-s + 4.01·38-s − 6.97·40-s + 2.29·41-s + ⋯ |
| L(s) = 1 | + 0.337·2-s − 0.886·4-s + 1.73·5-s + 0.377·7-s − 0.635·8-s + 0.584·10-s − 1.50·11-s − 0.113·13-s + 0.127·14-s + 0.671·16-s − 0.172·17-s + 1.93·19-s − 1.53·20-s − 0.508·22-s + 1.12·23-s + 2.00·25-s − 0.0381·26-s − 0.335·28-s − 1.51·29-s − 0.142·31-s + 0.862·32-s − 0.0580·34-s + 0.655·35-s + 1.08·37-s + 0.651·38-s − 1.10·40-s + 0.359·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.697100127\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.697100127\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 127 | \( 1 + T \) |
| good | 2 | \( 1 - 0.476T + 2T^{2} \) |
| 5 | \( 1 - 3.87T + 5T^{2} \) |
| 11 | \( 1 + 5.00T + 11T^{2} \) |
| 13 | \( 1 + 0.407T + 13T^{2} \) |
| 17 | \( 1 + 0.710T + 17T^{2} \) |
| 19 | \( 1 - 8.42T + 19T^{2} \) |
| 23 | \( 1 - 5.39T + 23T^{2} \) |
| 29 | \( 1 + 8.15T + 29T^{2} \) |
| 31 | \( 1 + 0.794T + 31T^{2} \) |
| 37 | \( 1 - 6.57T + 37T^{2} \) |
| 41 | \( 1 - 2.29T + 41T^{2} \) |
| 43 | \( 1 - 4.17T + 43T^{2} \) |
| 47 | \( 1 + 9.25T + 47T^{2} \) |
| 53 | \( 1 + 1.76T + 53T^{2} \) |
| 59 | \( 1 - 1.08T + 59T^{2} \) |
| 61 | \( 1 - 1.11T + 61T^{2} \) |
| 67 | \( 1 - 0.810T + 67T^{2} \) |
| 71 | \( 1 + 9.28T + 71T^{2} \) |
| 73 | \( 1 + 11.8T + 73T^{2} \) |
| 79 | \( 1 - 3.16T + 79T^{2} \) |
| 83 | \( 1 + 1.77T + 83T^{2} \) |
| 89 | \( 1 - 7.40T + 89T^{2} \) |
| 97 | \( 1 - 1.89T + 97T^{2} \) |
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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
−7.71562840023725822674107056717, −7.28209032385524158748754758539, −6.03519187467919411370623679078, −5.66558831127592986037891380957, −5.06890363247787203551058674200, −4.68510616543738787442445143537, −3.32707340925388440398617463067, −2.76687553901618981188757630449, −1.82507652591140837175336314764, −0.792957047784232780460323288938,
0.792957047784232780460323288938, 1.82507652591140837175336314764, 2.76687553901618981188757630449, 3.32707340925388440398617463067, 4.68510616543738787442445143537, 5.06890363247787203551058674200, 5.66558831127592986037891380957, 6.03519187467919411370623679078, 7.28209032385524158748754758539, 7.71562840023725822674107056717
