| L(s) = 1 | − 0.716·2-s − 1.48·4-s − 0.617·5-s − 7-s + 2.49·8-s + 0.442·10-s − 2.03·11-s − 2.26·13-s + 0.716·14-s + 1.18·16-s + 1.11·17-s + 7.92·19-s + 0.918·20-s + 1.46·22-s − 6.62·23-s − 4.61·25-s + 1.62·26-s + 1.48·28-s + 3.70·29-s − 1.50·31-s − 5.84·32-s − 0.800·34-s + 0.617·35-s + 4.29·37-s − 5.67·38-s − 1.54·40-s + 2.12·41-s + ⋯ |
| L(s) = 1 | − 0.506·2-s − 0.743·4-s − 0.276·5-s − 0.377·7-s + 0.883·8-s + 0.140·10-s − 0.614·11-s − 0.627·13-s + 0.191·14-s + 0.295·16-s + 0.271·17-s + 1.81·19-s + 0.205·20-s + 0.311·22-s − 1.38·23-s − 0.923·25-s + 0.318·26-s + 0.280·28-s + 0.687·29-s − 0.270·31-s − 1.03·32-s − 0.137·34-s + 0.104·35-s + 0.706·37-s − 0.921·38-s − 0.244·40-s + 0.331·41-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
| \(L(1)\) |
\(\approx\) |
\(0.6499241157\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6499241157\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.716T + 2T^{2} \) |
| 5 | \( 1 + 0.617T + 5T^{2} \) |
| 11 | \( 1 + 2.03T + 11T^{2} \) |
| 13 | \( 1 + 2.26T + 13T^{2} \) |
| 17 | \( 1 - 1.11T + 17T^{2} \) |
| 19 | \( 1 - 7.92T + 19T^{2} \) |
| 23 | \( 1 + 6.62T + 23T^{2} \) |
| 29 | \( 1 - 3.70T + 29T^{2} \) |
| 31 | \( 1 + 1.50T + 31T^{2} \) |
| 37 | \( 1 - 4.29T + 37T^{2} \) |
| 41 | \( 1 - 2.12T + 41T^{2} \) |
| 43 | \( 1 + 5.68T + 43T^{2} \) |
| 47 | \( 1 + 7.25T + 47T^{2} \) |
| 53 | \( 1 + 7.02T + 53T^{2} \) |
| 59 | \( 1 - 1.93T + 59T^{2} \) |
| 61 | \( 1 - 7.43T + 61T^{2} \) |
| 67 | \( 1 - 10.4T + 67T^{2} \) |
| 71 | \( 1 + 9.88T + 71T^{2} \) |
| 73 | \( 1 - 5.42T + 73T^{2} \) |
| 79 | \( 1 - 15.1T + 79T^{2} \) |
| 83 | \( 1 + 8.98T + 83T^{2} \) |
| 89 | \( 1 + 10.5T + 89T^{2} \) |
| 97 | \( 1 + 14.4T + 97T^{2} \) |
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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
−7.964554435243201615263684156270, −7.42564818351176992638748420567, −6.56588587469943482040651105619, −5.55446200846763591875259694328, −5.14582522966369208307741904232, −4.25069516167729427027341048869, −3.55938734294954694273710535886, −2.70726514472891922643651911875, −1.56381497886385656763821908846, −0.44423456251878473783349541239,
0.44423456251878473783349541239, 1.56381497886385656763821908846, 2.70726514472891922643651911875, 3.55938734294954694273710535886, 4.25069516167729427027341048869, 5.14582522966369208307741904232, 5.55446200846763591875259694328, 6.56588587469943482040651105619, 7.42564818351176992638748420567, 7.964554435243201615263684156270
