| L(s) = 1 | − 1.50·2-s + 0.265·4-s + 2.12·5-s + 7-s + 2.61·8-s − 3.20·10-s − 1.63·11-s + 2.77·13-s − 1.50·14-s − 4.46·16-s + 4.81·17-s + 1.82·19-s + 0.564·20-s + 2.45·22-s − 4.21·23-s − 0.471·25-s − 4.17·26-s + 0.265·28-s + 2.17·29-s − 5.01·31-s + 1.49·32-s − 7.24·34-s + 2.12·35-s − 4.80·37-s − 2.74·38-s + 5.55·40-s + 0.803·41-s + ⋯ |
| L(s) = 1 | − 1.06·2-s + 0.132·4-s + 0.951·5-s + 0.377·7-s + 0.923·8-s − 1.01·10-s − 0.491·11-s + 0.769·13-s − 0.402·14-s − 1.11·16-s + 1.16·17-s + 0.417·19-s + 0.126·20-s + 0.523·22-s − 0.879·23-s − 0.0943·25-s − 0.818·26-s + 0.0501·28-s + 0.403·29-s − 0.900·31-s + 0.263·32-s − 1.24·34-s + 0.359·35-s − 0.790·37-s − 0.444·38-s + 0.878·40-s + 0.125·41-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
| \(L(1)\) |
\(\approx\) |
\(1.416413103\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.416413103\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.50T + 2T^{2} \) |
| 5 | \( 1 - 2.12T + 5T^{2} \) |
| 11 | \( 1 + 1.63T + 11T^{2} \) |
| 13 | \( 1 - 2.77T + 13T^{2} \) |
| 17 | \( 1 - 4.81T + 17T^{2} \) |
| 19 | \( 1 - 1.82T + 19T^{2} \) |
| 23 | \( 1 + 4.21T + 23T^{2} \) |
| 29 | \( 1 - 2.17T + 29T^{2} \) |
| 31 | \( 1 + 5.01T + 31T^{2} \) |
| 37 | \( 1 + 4.80T + 37T^{2} \) |
| 41 | \( 1 - 0.803T + 41T^{2} \) |
| 43 | \( 1 - 11.2T + 43T^{2} \) |
| 47 | \( 1 - 7.13T + 47T^{2} \) |
| 53 | \( 1 + 7.35T + 53T^{2} \) |
| 59 | \( 1 + 1.99T + 59T^{2} \) |
| 61 | \( 1 - 1.64T + 61T^{2} \) |
| 67 | \( 1 - 10.7T + 67T^{2} \) |
| 71 | \( 1 + 4.48T + 71T^{2} \) |
| 73 | \( 1 - 5.53T + 73T^{2} \) |
| 79 | \( 1 + 0.583T + 79T^{2} \) |
| 83 | \( 1 - 4.31T + 83T^{2} \) |
| 89 | \( 1 - 0.157T + 89T^{2} \) |
| 97 | \( 1 - 2.33T + 97T^{2} \) |
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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
−7.75831781106462340414506271265, −7.58586460829543646333145116465, −6.50555835539637353190269964697, −5.69985985267654076602550002826, −5.28700787154591036858373317889, −4.29270075200245512087787607130, −3.45360576876460836639086762643, −2.29040659264339190471942499458, −1.59468485810115376658209330656, −0.73925716073992146419255707632,
0.73925716073992146419255707632, 1.59468485810115376658209330656, 2.29040659264339190471942499458, 3.45360576876460836639086762643, 4.29270075200245512087787607130, 5.28700787154591036858373317889, 5.69985985267654076602550002826, 6.50555835539637353190269964697, 7.58586460829543646333145116465, 7.75831781106462340414506271265
