L(s) = 1 | − 0.840·2-s − 1.29·4-s − 2.74·5-s + 7-s + 2.76·8-s + 2.30·10-s + 5.62·11-s − 3.65·13-s − 0.840·14-s + 0.257·16-s + 5.86·17-s − 2.05·19-s + 3.55·20-s − 4.72·22-s − 2.78·23-s + 2.54·25-s + 3.07·26-s − 1.29·28-s + 4.83·29-s − 6.87·31-s − 5.75·32-s − 4.93·34-s − 2.74·35-s − 8.71·37-s + 1.73·38-s − 7.60·40-s + 4.09·41-s + ⋯ |
L(s) = 1 | − 0.594·2-s − 0.646·4-s − 1.22·5-s + 0.377·7-s + 0.978·8-s + 0.730·10-s + 1.69·11-s − 1.01·13-s − 0.224·14-s + 0.0644·16-s + 1.42·17-s − 0.472·19-s + 0.794·20-s − 1.00·22-s − 0.580·23-s + 0.509·25-s + 0.602·26-s − 0.244·28-s + 0.897·29-s − 1.23·31-s − 1.01·32-s − 0.845·34-s − 0.464·35-s − 1.43·37-s + 0.280·38-s − 1.20·40-s + 0.640·41-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 | \( 1 + 0.840T + 2T^{2} \) |
| 5 | \( 1 + 2.74T + 5T^{2} \) |
| 11 | \( 1 - 5.62T + 11T^{2} \) |
| 13 | \( 1 + 3.65T + 13T^{2} \) |
| 17 | \( 1 - 5.86T + 17T^{2} \) |
| 19 | \( 1 + 2.05T + 19T^{2} \) |
| 23 | \( 1 + 2.78T + 23T^{2} \) |
| 29 | \( 1 - 4.83T + 29T^{2} \) |
| 31 | \( 1 + 6.87T + 31T^{2} \) |
| 37 | \( 1 + 8.71T + 37T^{2} \) |
| 41 | \( 1 - 4.09T + 41T^{2} \) |
| 43 | \( 1 + 2.48T + 43T^{2} \) |
| 47 | \( 1 + 12.6T + 47T^{2} \) |
| 53 | \( 1 + 4.92T + 53T^{2} \) |
| 59 | \( 1 - 9.41T + 59T^{2} \) |
| 61 | \( 1 - 4.96T + 61T^{2} \) |
| 67 | \( 1 - 4.57T + 67T^{2} \) |
| 71 | \( 1 + 2.13T + 71T^{2} \) |
| 73 | \( 1 - 9.59T + 73T^{2} \) |
| 79 | \( 1 + 10.0T + 79T^{2} \) |
| 83 | \( 1 - 5.06T + 83T^{2} \) |
| 89 | \( 1 + 2.73T + 89T^{2} \) |
| 97 | \( 1 - 14.0T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.52006084767446143608963827652, −7.16730507560840956847881734027, −6.23494777040785515352059560829, −5.15844508203612161793623743119, −4.64165892858926490958881529416, −3.76954604134175659866502237157, −3.50662786607944416646579748805, −1.92792248334769426393719457636, −1.03589665077963974341060406936, 0,
1.03589665077963974341060406936, 1.92792248334769426393719457636, 3.50662786607944416646579748805, 3.76954604134175659866502237157, 4.64165892858926490958881529416, 5.15844508203612161793623743119, 6.23494777040785515352059560829, 7.16730507560840956847881734027, 7.52006084767446143608963827652