| L(s) = 1 | − 0.959·2-s − 1.07·4-s − 2.40·5-s − 7-s + 2.95·8-s + 2.30·10-s − 4.25·11-s + 3.54·13-s + 0.959·14-s − 0.679·16-s − 0.302·17-s − 6.75·19-s + 2.59·20-s + 4.08·22-s + 6.51·23-s + 0.782·25-s − 3.40·26-s + 1.07·28-s + 3.90·29-s − 5.48·31-s − 5.25·32-s + 0.290·34-s + 2.40·35-s − 6.55·37-s + 6.48·38-s − 7.10·40-s + 2.17·41-s + ⋯ |
| L(s) = 1 | − 0.678·2-s − 0.539·4-s − 1.07·5-s − 0.377·7-s + 1.04·8-s + 0.729·10-s − 1.28·11-s + 0.982·13-s + 0.256·14-s − 0.169·16-s − 0.0733·17-s − 1.55·19-s + 0.579·20-s + 0.871·22-s + 1.35·23-s + 0.156·25-s − 0.667·26-s + 0.203·28-s + 0.725·29-s − 0.985·31-s − 0.929·32-s + 0.0497·34-s + 0.406·35-s − 1.07·37-s + 1.05·38-s − 1.12·40-s + 0.339·41-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 | \( 1 + 0.959T + 2T^{2} \) |
| 5 | \( 1 + 2.40T + 5T^{2} \) |
| 11 | \( 1 + 4.25T + 11T^{2} \) |
| 13 | \( 1 - 3.54T + 13T^{2} \) |
| 17 | \( 1 + 0.302T + 17T^{2} \) |
| 19 | \( 1 + 6.75T + 19T^{2} \) |
| 23 | \( 1 - 6.51T + 23T^{2} \) |
| 29 | \( 1 - 3.90T + 29T^{2} \) |
| 31 | \( 1 + 5.48T + 31T^{2} \) |
| 37 | \( 1 + 6.55T + 37T^{2} \) |
| 41 | \( 1 - 2.17T + 41T^{2} \) |
| 43 | \( 1 - 9.47T + 43T^{2} \) |
| 47 | \( 1 - 1.43T + 47T^{2} \) |
| 53 | \( 1 - 10.5T + 53T^{2} \) |
| 59 | \( 1 + 7.97T + 59T^{2} \) |
| 61 | \( 1 - 1.05T + 61T^{2} \) |
| 67 | \( 1 + 4.72T + 67T^{2} \) |
| 71 | \( 1 - 6.25T + 71T^{2} \) |
| 73 | \( 1 + 11.4T + 73T^{2} \) |
| 79 | \( 1 - 4.55T + 79T^{2} \) |
| 83 | \( 1 - 10.1T + 83T^{2} \) |
| 89 | \( 1 - 8.24T + 89T^{2} \) |
| 97 | \( 1 - 1.74T + 97T^{2} \) |
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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
−7.59970482167619005118940600617, −7.12616224287042355910778609578, −6.17027274396159190964288497286, −5.32126214012925169024880773875, −4.55622309592903232639882635400, −3.93909388895491173980303612203, −3.20728842791737800991915013102, −2.14174786823942383315208645829, −0.847466171499621743089099854139, 0,
0.847466171499621743089099854139, 2.14174786823942383315208645829, 3.20728842791737800991915013102, 3.93909388895491173980303612203, 4.55622309592903232639882635400, 5.32126214012925169024880773875, 6.17027274396159190964288497286, 7.12616224287042355910778609578, 7.59970482167619005118940600617
