| L(s) = 1 | − 2.73·2-s − 3.10·3-s + 5.45·4-s − 3.71·5-s + 8.47·6-s + 1.88·7-s − 9.43·8-s + 6.64·9-s + 10.1·10-s + 0.489·11-s − 16.9·12-s + 5.48·13-s − 5.15·14-s + 11.5·15-s + 14.8·16-s + 17-s − 18.1·18-s − 0.332·19-s − 20.2·20-s − 5.86·21-s − 1.33·22-s + 8.82·23-s + 29.2·24-s + 8.79·25-s − 14.9·26-s − 11.3·27-s + 10.2·28-s + ⋯ |
| L(s) = 1 | − 1.93·2-s − 1.79·3-s + 2.72·4-s − 1.66·5-s + 3.46·6-s + 0.713·7-s − 3.33·8-s + 2.21·9-s + 3.20·10-s + 0.147·11-s − 4.88·12-s + 1.52·13-s − 1.37·14-s + 2.97·15-s + 3.70·16-s + 0.242·17-s − 4.27·18-s − 0.0763·19-s − 4.52·20-s − 1.27·21-s − 0.284·22-s + 1.83·23-s + 5.97·24-s + 1.75·25-s − 2.93·26-s − 2.17·27-s + 1.94·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 17 | \( 1 - T \) |
| 353 | \( 1 - T \) |
| good | 2 | \( 1 + 2.73T + 2T^{2} \) |
| 3 | \( 1 + 3.10T + 3T^{2} \) |
| 5 | \( 1 + 3.71T + 5T^{2} \) |
| 7 | \( 1 - 1.88T + 7T^{2} \) |
| 11 | \( 1 - 0.489T + 11T^{2} \) |
| 13 | \( 1 - 5.48T + 13T^{2} \) |
| 19 | \( 1 + 0.332T + 19T^{2} \) |
| 23 | \( 1 - 8.82T + 23T^{2} \) |
| 29 | \( 1 - 0.482T + 29T^{2} \) |
| 31 | \( 1 + 5.74T + 31T^{2} \) |
| 37 | \( 1 - 8.83T + 37T^{2} \) |
| 41 | \( 1 + 4.82T + 41T^{2} \) |
| 43 | \( 1 - 4.70T + 43T^{2} \) |
| 47 | \( 1 + 9.37T + 47T^{2} \) |
| 53 | \( 1 - 6.17T + 53T^{2} \) |
| 59 | \( 1 - 1.46T + 59T^{2} \) |
| 61 | \( 1 + 6.01T + 61T^{2} \) |
| 67 | \( 1 - 6.95T + 67T^{2} \) |
| 71 | \( 1 + 10.4T + 71T^{2} \) |
| 73 | \( 1 + 14.5T + 73T^{2} \) |
| 79 | \( 1 - 3.18T + 79T^{2} \) |
| 83 | \( 1 + 7.61T + 83T^{2} \) |
| 89 | \( 1 + 10.6T + 89T^{2} \) |
| 97 | \( 1 + 13.5T + 97T^{2} \) |
| 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
−7.63457380769352174770250060502, −7.25913507493000287289461459440, −6.57554392058906956583091302488, −5.92016482768595465922885800012, −5.00751983053737036844426136748, −4.07165350121672826451213116465, −3.12129026741180416563771620103, −1.42955888688563344826815692305, −0.966256976494360434713748111863, 0,
0.966256976494360434713748111863, 1.42955888688563344826815692305, 3.12129026741180416563771620103, 4.07165350121672826451213116465, 5.00751983053737036844426136748, 5.92016482768595465922885800012, 6.57554392058906956583091302488, 7.25913507493000287289461459440, 7.63457380769352174770250060502
