| L(s) = 1 | + 0.678·2-s − 1.53·4-s − 2.84·5-s − 7-s − 2.40·8-s − 1.93·10-s − 1.95·11-s − 0.416·13-s − 0.678·14-s + 1.44·16-s + 6.48·17-s − 6.07·19-s + 4.37·20-s − 1.32·22-s + 4.22·23-s + 3.09·25-s − 0.282·26-s + 1.53·28-s + 10.6·29-s − 8.61·31-s + 5.78·32-s + 4.40·34-s + 2.84·35-s + 0.268·37-s − 4.12·38-s + 6.83·40-s + 1.71·41-s + ⋯ |
| L(s) = 1 | + 0.479·2-s − 0.769·4-s − 1.27·5-s − 0.377·7-s − 0.849·8-s − 0.610·10-s − 0.590·11-s − 0.115·13-s − 0.181·14-s + 0.362·16-s + 1.57·17-s − 1.39·19-s + 0.979·20-s − 0.283·22-s + 0.880·23-s + 0.618·25-s − 0.0554·26-s + 0.290·28-s + 1.98·29-s − 1.54·31-s + 1.02·32-s + 0.755·34-s + 0.480·35-s + 0.0441·37-s − 0.668·38-s + 1.08·40-s + 0.267·41-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.678T + 2T^{2} \) |
| 5 | \( 1 + 2.84T + 5T^{2} \) |
| 11 | \( 1 + 1.95T + 11T^{2} \) |
| 13 | \( 1 + 0.416T + 13T^{2} \) |
| 17 | \( 1 - 6.48T + 17T^{2} \) |
| 19 | \( 1 + 6.07T + 19T^{2} \) |
| 23 | \( 1 - 4.22T + 23T^{2} \) |
| 29 | \( 1 - 10.6T + 29T^{2} \) |
| 31 | \( 1 + 8.61T + 31T^{2} \) |
| 37 | \( 1 - 0.268T + 37T^{2} \) |
| 41 | \( 1 - 1.71T + 41T^{2} \) |
| 43 | \( 1 - 7.53T + 43T^{2} \) |
| 47 | \( 1 + 3.95T + 47T^{2} \) |
| 53 | \( 1 + 0.195T + 53T^{2} \) |
| 59 | \( 1 + 0.232T + 59T^{2} \) |
| 61 | \( 1 + 2.32T + 61T^{2} \) |
| 67 | \( 1 + 5.65T + 67T^{2} \) |
| 71 | \( 1 - 13.5T + 71T^{2} \) |
| 73 | \( 1 - 10.5T + 73T^{2} \) |
| 79 | \( 1 - 9.40T + 79T^{2} \) |
| 83 | \( 1 + 12.8T + 83T^{2} \) |
| 89 | \( 1 + 8.15T + 89T^{2} \) |
| 97 | \( 1 - 0.572T + 97T^{2} \) |
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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
−7.63903805851150887791856978973, −6.79821824132378350641849197239, −5.99144008370147552491950606116, −5.21926960348501030800494588990, −4.62167732488491213109549446930, −3.88693113693961127658754819597, −3.35192240950817271602455103601, −2.58877065759910047158906653668, −0.944961288736862878707725065519, 0,
0.944961288736862878707725065519, 2.58877065759910047158906653668, 3.35192240950817271602455103601, 3.88693113693961127658754819597, 4.62167732488491213109549446930, 5.21926960348501030800494588990, 5.99144008370147552491950606116, 6.79821824132378350641849197239, 7.63903805851150887791856978973
