| L(s) = 1 | + 2.71·2-s + 5.38·4-s + 1.23·5-s − 7-s + 9.18·8-s + 3.35·10-s − 0.379·11-s − 5.61·13-s − 2.71·14-s + 14.1·16-s + 3.36·17-s + 7.76·19-s + 6.65·20-s − 1.02·22-s − 1.82·23-s − 3.47·25-s − 15.2·26-s − 5.38·28-s + 5.71·29-s + 5.94·31-s + 20.1·32-s + 9.13·34-s − 1.23·35-s + 3.72·37-s + 21.1·38-s + 11.3·40-s − 2.69·41-s + ⋯ |
| L(s) = 1 | + 1.92·2-s + 2.69·4-s + 0.553·5-s − 0.377·7-s + 3.24·8-s + 1.06·10-s − 0.114·11-s − 1.55·13-s − 0.726·14-s + 3.54·16-s + 0.815·17-s + 1.78·19-s + 1.48·20-s − 0.219·22-s − 0.381·23-s − 0.694·25-s − 2.99·26-s − 1.01·28-s + 1.06·29-s + 1.06·31-s + 3.57·32-s + 1.56·34-s − 0.209·35-s + 0.612·37-s + 3.42·38-s + 1.79·40-s − 0.420·41-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
| \(L(1)\) |
\(\approx\) |
\(8.656140481\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.656140481\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.71T + 2T^{2} \) |
| 5 | \( 1 - 1.23T + 5T^{2} \) |
| 11 | \( 1 + 0.379T + 11T^{2} \) |
| 13 | \( 1 + 5.61T + 13T^{2} \) |
| 17 | \( 1 - 3.36T + 17T^{2} \) |
| 19 | \( 1 - 7.76T + 19T^{2} \) |
| 23 | \( 1 + 1.82T + 23T^{2} \) |
| 29 | \( 1 - 5.71T + 29T^{2} \) |
| 31 | \( 1 - 5.94T + 31T^{2} \) |
| 37 | \( 1 - 3.72T + 37T^{2} \) |
| 41 | \( 1 + 2.69T + 41T^{2} \) |
| 43 | \( 1 + 0.308T + 43T^{2} \) |
| 47 | \( 1 - 2.68T + 47T^{2} \) |
| 53 | \( 1 + 4.51T + 53T^{2} \) |
| 59 | \( 1 - 9.48T + 59T^{2} \) |
| 61 | \( 1 - 13.1T + 61T^{2} \) |
| 67 | \( 1 + 5.03T + 67T^{2} \) |
| 71 | \( 1 - 6.46T + 71T^{2} \) |
| 73 | \( 1 - 2.19T + 73T^{2} \) |
| 79 | \( 1 + 11.5T + 79T^{2} \) |
| 83 | \( 1 - 4.17T + 83T^{2} \) |
| 89 | \( 1 - 16.5T + 89T^{2} \) |
| 97 | \( 1 + 7.99T + 97T^{2} \) |
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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
−7.55117587157920993340727282258, −6.86917434438818810213737715180, −6.27777672780414160476567062495, −5.40982948132628374637039094320, −5.22852852810238910070334050678, −4.37760211809622865357802700110, −3.54881815963795739178513835368, −2.79143803506375424935300801746, −2.33262699505920546353953687165, −1.15033207020031661153216898324,
1.15033207020031661153216898324, 2.33262699505920546353953687165, 2.79143803506375424935300801746, 3.54881815963795739178513835368, 4.37760211809622865357802700110, 5.22852852810238910070334050678, 5.40982948132628374637039094320, 6.27777672780414160476567062495, 6.86917434438818810213737715180, 7.55117587157920993340727282258
