| L(s) = 1 | + 1.09·2-s − 0.803·4-s − 2.53·5-s − 7-s − 3.06·8-s − 2.77·10-s − 1.22·11-s − 1.28·13-s − 1.09·14-s − 1.74·16-s + 2.38·17-s − 2.44·19-s + 2.03·20-s − 1.34·22-s − 2.08·23-s + 1.44·25-s − 1.40·26-s + 0.803·28-s − 4.47·29-s − 3.34·31-s + 4.21·32-s + 2.60·34-s + 2.53·35-s − 8.53·37-s − 2.67·38-s + 7.78·40-s − 3.94·41-s + ⋯ |
| L(s) = 1 | + 0.773·2-s − 0.401·4-s − 1.13·5-s − 0.377·7-s − 1.08·8-s − 0.878·10-s − 0.369·11-s − 0.355·13-s − 0.292·14-s − 0.437·16-s + 0.577·17-s − 0.560·19-s + 0.455·20-s − 0.286·22-s − 0.435·23-s + 0.289·25-s − 0.275·26-s + 0.151·28-s − 0.831·29-s − 0.600·31-s + 0.745·32-s + 0.446·34-s + 0.429·35-s − 1.40·37-s − 0.433·38-s + 1.23·40-s − 0.616·41-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
| \(L(1)\) |
\(\approx\) |
\(0.5546936015\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5546936015\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.09T + 2T^{2} \) |
| 5 | \( 1 + 2.53T + 5T^{2} \) |
| 11 | \( 1 + 1.22T + 11T^{2} \) |
| 13 | \( 1 + 1.28T + 13T^{2} \) |
| 17 | \( 1 - 2.38T + 17T^{2} \) |
| 19 | \( 1 + 2.44T + 19T^{2} \) |
| 23 | \( 1 + 2.08T + 23T^{2} \) |
| 29 | \( 1 + 4.47T + 29T^{2} \) |
| 31 | \( 1 + 3.34T + 31T^{2} \) |
| 37 | \( 1 + 8.53T + 37T^{2} \) |
| 41 | \( 1 + 3.94T + 41T^{2} \) |
| 43 | \( 1 + 9.49T + 43T^{2} \) |
| 47 | \( 1 + 4.48T + 47T^{2} \) |
| 53 | \( 1 - 10.1T + 53T^{2} \) |
| 59 | \( 1 + 0.886T + 59T^{2} \) |
| 61 | \( 1 + 5.46T + 61T^{2} \) |
| 67 | \( 1 - 11.3T + 67T^{2} \) |
| 71 | \( 1 + 11.2T + 71T^{2} \) |
| 73 | \( 1 + 11.5T + 73T^{2} \) |
| 79 | \( 1 - 2.25T + 79T^{2} \) |
| 83 | \( 1 + 4.64T + 83T^{2} \) |
| 89 | \( 1 + 9.85T + 89T^{2} \) |
| 97 | \( 1 - 14.9T + 97T^{2} \) |
| 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
−7.73840066802822649454352556056, −7.19072596843075054677365139796, −6.34793916563211222631118750929, −5.56276113916911524124247209091, −4.97388547753617819306932060427, −4.22404805937612321259545525704, −3.56700948211584659681836972314, −3.13841667241424881958608542292, −1.92596936113355722641966400071, −0.31230906647531436880550915440,
0.31230906647531436880550915440, 1.92596936113355722641966400071, 3.13841667241424881958608542292, 3.56700948211584659681836972314, 4.22404805937612321259545525704, 4.97388547753617819306932060427, 5.56276113916911524124247209091, 6.34793916563211222631118750929, 7.19072596843075054677365139796, 7.73840066802822649454352556056
