L(s) = 1 | + 2.72·2-s + 5.44·4-s − 0.297·5-s − 7-s + 9.39·8-s − 0.811·10-s − 1.00·11-s + 6.53·13-s − 2.72·14-s + 14.7·16-s − 0.490·17-s + 1.20·19-s − 1.61·20-s − 2.74·22-s + 5.71·23-s − 4.91·25-s + 17.8·26-s − 5.44·28-s − 2.54·29-s + 1.43·31-s + 21.4·32-s − 1.33·34-s + 0.297·35-s − 4.61·37-s + 3.27·38-s − 2.79·40-s + 7.46·41-s + ⋯ |
L(s) = 1 | + 1.92·2-s + 2.72·4-s − 0.133·5-s − 0.377·7-s + 3.32·8-s − 0.256·10-s − 0.303·11-s + 1.81·13-s − 0.729·14-s + 3.68·16-s − 0.118·17-s + 0.275·19-s − 0.362·20-s − 0.585·22-s + 1.19·23-s − 0.982·25-s + 3.49·26-s − 1.02·28-s − 0.472·29-s + 0.256·31-s + 3.78·32-s − 0.229·34-s + 0.0502·35-s − 0.759·37-s + 0.531·38-s − 0.441·40-s + 1.16·41-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
\(L(1)\) |
\(\approx\) |
\(8.346891071\) |
\(L(\frac12)\) |
\(\approx\) |
\(8.346891071\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.72T + 2T^{2} \) |
| 5 | \( 1 + 0.297T + 5T^{2} \) |
| 11 | \( 1 + 1.00T + 11T^{2} \) |
| 13 | \( 1 - 6.53T + 13T^{2} \) |
| 17 | \( 1 + 0.490T + 17T^{2} \) |
| 19 | \( 1 - 1.20T + 19T^{2} \) |
| 23 | \( 1 - 5.71T + 23T^{2} \) |
| 29 | \( 1 + 2.54T + 29T^{2} \) |
| 31 | \( 1 - 1.43T + 31T^{2} \) |
| 37 | \( 1 + 4.61T + 37T^{2} \) |
| 41 | \( 1 - 7.46T + 41T^{2} \) |
| 43 | \( 1 - 2.51T + 43T^{2} \) |
| 47 | \( 1 + 5.66T + 47T^{2} \) |
| 53 | \( 1 + 1.09T + 53T^{2} \) |
| 59 | \( 1 - 2.52T + 59T^{2} \) |
| 61 | \( 1 - 6.71T + 61T^{2} \) |
| 67 | \( 1 - 4.35T + 67T^{2} \) |
| 71 | \( 1 + 9.33T + 71T^{2} \) |
| 73 | \( 1 + 4.25T + 73T^{2} \) |
| 79 | \( 1 - 5.38T + 79T^{2} \) |
| 83 | \( 1 - 2.93T + 83T^{2} \) |
| 89 | \( 1 + 10.4T + 89T^{2} \) |
| 97 | \( 1 - 2.21T + 97T^{2} \) |
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
−7.49795502979784506341052671261, −6.87712286332996599910727151754, −6.16336223627607650655120514085, −5.74559889973396710914038747772, −5.02420725619504270913853969710, −4.20179585621242335571101184499, −3.59521045613249633321821958305, −3.07916535785199760188350208050, −2.13188036460854583979269878181, −1.14790023615198122733707773993,
1.14790023615198122733707773993, 2.13188036460854583979269878181, 3.07916535785199760188350208050, 3.59521045613249633321821958305, 4.20179585621242335571101184499, 5.02420725619504270913853969710, 5.74559889973396710914038747772, 6.16336223627607650655120514085, 6.87712286332996599910727151754, 7.49795502979784506341052671261