| L(s) = 1 | + 2.59·2-s + 4.73·4-s + 1.63·5-s + 7-s + 7.11·8-s + 4.23·10-s − 4.79·11-s + 0.263·13-s + 2.59·14-s + 8.98·16-s + 5.12·17-s + 5.93·19-s + 7.73·20-s − 12.4·22-s + 4.10·23-s − 2.33·25-s + 0.684·26-s + 4.73·28-s − 0.459·29-s − 9.24·31-s + 9.09·32-s + 13.2·34-s + 1.63·35-s + 6.64·37-s + 15.4·38-s + 11.6·40-s + 3.43·41-s + ⋯ |
| L(s) = 1 | + 1.83·2-s + 2.36·4-s + 0.730·5-s + 0.377·7-s + 2.51·8-s + 1.34·10-s − 1.44·11-s + 0.0731·13-s + 0.693·14-s + 2.24·16-s + 1.24·17-s + 1.36·19-s + 1.72·20-s − 2.65·22-s + 0.855·23-s − 0.467·25-s + 0.134·26-s + 0.895·28-s − 0.0853·29-s − 1.66·31-s + 1.60·32-s + 2.27·34-s + 0.275·35-s + 1.09·37-s + 2.49·38-s + 1.83·40-s + 0.536·41-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
| \(L(1)\) |
\(\approx\) |
\(8.671078905\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.671078905\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.59T + 2T^{2} \) |
| 5 | \( 1 - 1.63T + 5T^{2} \) |
| 11 | \( 1 + 4.79T + 11T^{2} \) |
| 13 | \( 1 - 0.263T + 13T^{2} \) |
| 17 | \( 1 - 5.12T + 17T^{2} \) |
| 19 | \( 1 - 5.93T + 19T^{2} \) |
| 23 | \( 1 - 4.10T + 23T^{2} \) |
| 29 | \( 1 + 0.459T + 29T^{2} \) |
| 31 | \( 1 + 9.24T + 31T^{2} \) |
| 37 | \( 1 - 6.64T + 37T^{2} \) |
| 41 | \( 1 - 3.43T + 41T^{2} \) |
| 43 | \( 1 - 2.33T + 43T^{2} \) |
| 47 | \( 1 - 5.69T + 47T^{2} \) |
| 53 | \( 1 + 2.55T + 53T^{2} \) |
| 59 | \( 1 - 13.3T + 59T^{2} \) |
| 61 | \( 1 - 10.9T + 61T^{2} \) |
| 67 | \( 1 - 1.34T + 67T^{2} \) |
| 71 | \( 1 + 5.05T + 71T^{2} \) |
| 73 | \( 1 + 16.4T + 73T^{2} \) |
| 79 | \( 1 + 0.111T + 79T^{2} \) |
| 83 | \( 1 + 6.65T + 83T^{2} \) |
| 89 | \( 1 + 11.2T + 89T^{2} \) |
| 97 | \( 1 + 5.51T + 97T^{2} \) |
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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
−7.36043319725588326034376182068, −7.22716287872792873403923710558, −5.93601460531352146363187943585, −5.50405444436146237381237586860, −5.31579176656238177351670664277, −4.39148999076527276159056265080, −3.52372078539257202478194789020, −2.84145987745374769042751815387, −2.21164548132781392140877729912, −1.18933920074968271592727519936,
1.18933920074968271592727519936, 2.21164548132781392140877729912, 2.84145987745374769042751815387, 3.52372078539257202478194789020, 4.39148999076527276159056265080, 5.31579176656238177351670664277, 5.50405444436146237381237586860, 5.93601460531352146363187943585, 7.22716287872792873403923710558, 7.36043319725588326034376182068
