L(s) = 1 | − 0.291·2-s − 1.91·4-s − 1.20·7-s + 1.14·8-s − 3.65·11-s + 0.859·13-s + 0.351·14-s + 3.49·16-s − 6.72·17-s − 1.51·19-s + 1.06·22-s + 23-s − 0.250·26-s + 2.31·28-s − 0.548·29-s − 5.99·31-s − 3.30·32-s + 1.95·34-s + 2.04·37-s + 0.442·38-s + 7.14·41-s − 10.0·43-s + 7.00·44-s − 0.291·46-s + 9.17·47-s − 5.54·49-s − 1.64·52-s + ⋯ |
L(s) = 1 | − 0.206·2-s − 0.957·4-s − 0.456·7-s + 0.403·8-s − 1.10·11-s + 0.238·13-s + 0.0939·14-s + 0.874·16-s − 1.63·17-s − 0.348·19-s + 0.227·22-s + 0.208·23-s − 0.0491·26-s + 0.436·28-s − 0.101·29-s − 1.07·31-s − 0.583·32-s + 0.335·34-s + 0.335·37-s + 0.0717·38-s + 1.11·41-s − 1.53·43-s + 1.05·44-s − 0.0429·46-s + 1.33·47-s − 0.792·49-s − 0.228·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5826024471\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5826024471\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 - T \) |
good | 2 | \( 1 + 0.291T + 2T^{2} \) |
| 7 | \( 1 + 1.20T + 7T^{2} \) |
| 11 | \( 1 + 3.65T + 11T^{2} \) |
| 13 | \( 1 - 0.859T + 13T^{2} \) |
| 17 | \( 1 + 6.72T + 17T^{2} \) |
| 19 | \( 1 + 1.51T + 19T^{2} \) |
| 29 | \( 1 + 0.548T + 29T^{2} \) |
| 31 | \( 1 + 5.99T + 31T^{2} \) |
| 37 | \( 1 - 2.04T + 37T^{2} \) |
| 41 | \( 1 - 7.14T + 41T^{2} \) |
| 43 | \( 1 + 10.0T + 43T^{2} \) |
| 47 | \( 1 - 9.17T + 47T^{2} \) |
| 53 | \( 1 + 5.37T + 53T^{2} \) |
| 59 | \( 1 - 0.582T + 59T^{2} \) |
| 61 | \( 1 + 8.83T + 61T^{2} \) |
| 67 | \( 1 + 3.20T + 67T^{2} \) |
| 71 | \( 1 - 12.5T + 71T^{2} \) |
| 73 | \( 1 - 8.62T + 73T^{2} \) |
| 79 | \( 1 - 0.0700T + 79T^{2} \) |
| 83 | \( 1 + 6.74T + 83T^{2} \) |
| 89 | \( 1 + 4.96T + 89T^{2} \) |
| 97 | \( 1 - 11.3T + 97T^{2} \) |
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
−8.265380498381057434081435837343, −7.65881523037698066034042688728, −6.80847228358497630884499801041, −6.01933252730171983222627524790, −5.20630917378622828232158418366, −4.55662040775403515455615159688, −3.80779796086053480133206655023, −2.87108258844509845602722193555, −1.87109722173264211862830164592, −0.41550311587270894710531292561,
0.41550311587270894710531292561, 1.87109722173264211862830164592, 2.87108258844509845602722193555, 3.80779796086053480133206655023, 4.55662040775403515455615159688, 5.20630917378622828232158418366, 6.01933252730171983222627524790, 6.80847228358497630884499801041, 7.65881523037698066034042688728, 8.265380498381057434081435837343