| L(s) = 1 | + 0.487·2-s − 1.76·4-s − 0.708·5-s − 7-s − 1.83·8-s − 0.345·10-s − 4.44·11-s − 4.01·13-s − 0.487·14-s + 2.63·16-s − 1.16·17-s + 0.231·19-s + 1.24·20-s − 2.16·22-s − 6.97·23-s − 4.49·25-s − 1.95·26-s + 1.76·28-s − 0.276·29-s − 6.86·31-s + 4.94·32-s − 0.568·34-s + 0.708·35-s − 3.38·37-s + 0.112·38-s + 1.29·40-s − 7.55·41-s + ⋯ |
| L(s) = 1 | + 0.344·2-s − 0.881·4-s − 0.316·5-s − 0.377·7-s − 0.648·8-s − 0.109·10-s − 1.34·11-s − 1.11·13-s − 0.130·14-s + 0.657·16-s − 0.282·17-s + 0.0530·19-s + 0.279·20-s − 0.461·22-s − 1.45·23-s − 0.899·25-s − 0.384·26-s + 0.333·28-s − 0.0513·29-s − 1.23·31-s + 0.874·32-s − 0.0974·34-s + 0.119·35-s − 0.556·37-s + 0.0182·38-s + 0.205·40-s − 1.18·41-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
| \(L(1)\) |
\(\approx\) |
\(0.1553003462\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1553003462\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.487T + 2T^{2} \) |
| 5 | \( 1 + 0.708T + 5T^{2} \) |
| 11 | \( 1 + 4.44T + 11T^{2} \) |
| 13 | \( 1 + 4.01T + 13T^{2} \) |
| 17 | \( 1 + 1.16T + 17T^{2} \) |
| 19 | \( 1 - 0.231T + 19T^{2} \) |
| 23 | \( 1 + 6.97T + 23T^{2} \) |
| 29 | \( 1 + 0.276T + 29T^{2} \) |
| 31 | \( 1 + 6.86T + 31T^{2} \) |
| 37 | \( 1 + 3.38T + 37T^{2} \) |
| 41 | \( 1 + 7.55T + 41T^{2} \) |
| 43 | \( 1 - 5.27T + 43T^{2} \) |
| 47 | \( 1 - 5.93T + 47T^{2} \) |
| 53 | \( 1 + 10.2T + 53T^{2} \) |
| 59 | \( 1 + 9.69T + 59T^{2} \) |
| 61 | \( 1 - 9.65T + 61T^{2} \) |
| 67 | \( 1 + 5.37T + 67T^{2} \) |
| 71 | \( 1 + 2.44T + 71T^{2} \) |
| 73 | \( 1 + 10.1T + 73T^{2} \) |
| 79 | \( 1 - 9.67T + 79T^{2} \) |
| 83 | \( 1 - 4.65T + 83T^{2} \) |
| 89 | \( 1 + 2.71T + 89T^{2} \) |
| 97 | \( 1 - 5.35T + 97T^{2} \) |
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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
−7.75121781433264124520461079227, −7.38006700633509378225544367323, −6.27838371841547491914472015361, −5.57747046989178593676362138479, −5.05196832367484970232625374754, −4.30236941748194666720731106777, −3.62729734448503737746046608542, −2.80823110101465100583730077380, −1.93084150446095444250231375486, −0.17380846836805912723500525291,
0.17380846836805912723500525291, 1.93084150446095444250231375486, 2.80823110101465100583730077380, 3.62729734448503737746046608542, 4.30236941748194666720731106777, 5.05196832367484970232625374754, 5.57747046989178593676362138479, 6.27838371841547491914472015361, 7.38006700633509378225544367323, 7.75121781433264124520461079227
