| L(s) = 1 | + 1.24·2-s − 0.448·4-s + 2.43·5-s − 7-s − 3.04·8-s + 3.03·10-s + 4.22·11-s − 5.67·13-s − 1.24·14-s − 2.90·16-s + 1.72·17-s − 2.50·19-s − 1.09·20-s + 5.26·22-s + 3.83·23-s + 0.922·25-s − 7.07·26-s + 0.448·28-s − 5.96·29-s − 5.47·31-s + 2.48·32-s + 2.15·34-s − 2.43·35-s + 6.12·37-s − 3.11·38-s − 7.42·40-s + 4.31·41-s + ⋯ |
| L(s) = 1 | + 0.880·2-s − 0.224·4-s + 1.08·5-s − 0.377·7-s − 1.07·8-s + 0.958·10-s + 1.27·11-s − 1.57·13-s − 0.332·14-s − 0.725·16-s + 0.419·17-s − 0.574·19-s − 0.244·20-s + 1.12·22-s + 0.799·23-s + 0.184·25-s − 1.38·26-s + 0.0848·28-s − 1.10·29-s − 0.983·31-s + 0.439·32-s + 0.369·34-s − 0.411·35-s + 1.00·37-s − 0.505·38-s − 1.17·40-s + 0.673·41-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
| \(L(1)\) |
\(\approx\) |
\(3.015706436\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.015706436\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.24T + 2T^{2} \) |
| 5 | \( 1 - 2.43T + 5T^{2} \) |
| 11 | \( 1 - 4.22T + 11T^{2} \) |
| 13 | \( 1 + 5.67T + 13T^{2} \) |
| 17 | \( 1 - 1.72T + 17T^{2} \) |
| 19 | \( 1 + 2.50T + 19T^{2} \) |
| 23 | \( 1 - 3.83T + 23T^{2} \) |
| 29 | \( 1 + 5.96T + 29T^{2} \) |
| 31 | \( 1 + 5.47T + 31T^{2} \) |
| 37 | \( 1 - 6.12T + 37T^{2} \) |
| 41 | \( 1 - 4.31T + 41T^{2} \) |
| 43 | \( 1 - 4.93T + 43T^{2} \) |
| 47 | \( 1 - 10.9T + 47T^{2} \) |
| 53 | \( 1 - 4.00T + 53T^{2} \) |
| 59 | \( 1 - 14.3T + 59T^{2} \) |
| 61 | \( 1 - 3.21T + 61T^{2} \) |
| 67 | \( 1 - 7.39T + 67T^{2} \) |
| 71 | \( 1 + 7.63T + 71T^{2} \) |
| 73 | \( 1 - 7.53T + 73T^{2} \) |
| 79 | \( 1 + 15.4T + 79T^{2} \) |
| 83 | \( 1 - 12.7T + 83T^{2} \) |
| 89 | \( 1 + 3.88T + 89T^{2} \) |
| 97 | \( 1 - 4.24T + 97T^{2} \) |
| 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
−7.58153919251601519739007924751, −6.98413969065088785960543862893, −6.21647028103479911076753808221, −5.65728246622801530040385331761, −5.15349395962863142525189894684, −4.22881152100991969465503716086, −3.73493472793321746066874543634, −2.67384770425729897336614286931, −2.08458781769562858850110191793, −0.74183798428473109598441462606,
0.74183798428473109598441462606, 2.08458781769562858850110191793, 2.67384770425729897336614286931, 3.73493472793321746066874543634, 4.22881152100991969465503716086, 5.15349395962863142525189894684, 5.65728246622801530040385331761, 6.21647028103479911076753808221, 6.98413969065088785960543862893, 7.58153919251601519739007924751
