| L(s) = 1 | + i·5-s + 3.88·7-s − 5.08i·11-s + 3.88i·13-s − 4.22i·17-s + (1.78 − 3.97i)19-s + 5.40i·23-s − 25-s − 5.63·29-s − 5.88i·31-s + 3.88i·35-s − 8.10i·37-s − 6.49·41-s + 4.18·43-s − 12.1i·47-s + ⋯ |
| L(s) = 1 | + 0.447i·5-s + 1.46·7-s − 1.53i·11-s + 1.07i·13-s − 1.02i·17-s + (0.410 − 0.912i)19-s + 1.12i·23-s − 0.200·25-s − 1.04·29-s − 1.05i·31-s + 0.656i·35-s − 1.33i·37-s − 1.01·41-s + 0.637·43-s − 1.76i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.191 + 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.191 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.007252243\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.007252243\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 19 | \( 1 + (-1.78 + 3.97i)T \) |
| good | 7 | \( 1 - 3.88T + 7T^{2} \) |
| 11 | \( 1 + 5.08iT - 11T^{2} \) |
| 13 | \( 1 - 3.88iT - 13T^{2} \) |
| 17 | \( 1 + 4.22iT - 17T^{2} \) |
| 23 | \( 1 - 5.40iT - 23T^{2} \) |
| 29 | \( 1 + 5.63T + 29T^{2} \) |
| 31 | \( 1 + 5.88iT - 31T^{2} \) |
| 37 | \( 1 + 8.10iT - 37T^{2} \) |
| 41 | \( 1 + 6.49T + 41T^{2} \) |
| 43 | \( 1 - 4.18T + 43T^{2} \) |
| 47 | \( 1 + 12.1iT - 47T^{2} \) |
| 53 | \( 1 - 2.99T + 53T^{2} \) |
| 59 | \( 1 + 0.868T + 59T^{2} \) |
| 61 | \( 1 + 2.72T + 61T^{2} \) |
| 67 | \( 1 + 13.3iT - 67T^{2} \) |
| 71 | \( 1 - 3.68T + 71T^{2} \) |
| 73 | \( 1 + 3.19T + 73T^{2} \) |
| 79 | \( 1 - 9.49iT - 79T^{2} \) |
| 83 | \( 1 + 8.49iT - 83T^{2} \) |
| 89 | \( 1 + 10.8T + 89T^{2} \) |
| 97 | \( 1 - 0.856iT - 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.64469090833861159871366435995, −7.32169938013000843467778314462, −6.44390543513414604509483172759, −5.50793680658295110336144862565, −5.15696682519370010735554229829, −4.13922541736976302330697108910, −3.47007967754350658412691036466, −2.45003738538706181389386184223, −1.66427940631266967104065399108, −0.48543425141631370689351254368,
1.29465340253744837262668343304, 1.76005732489314773424876297752, 2.83422031864189027970050122509, 4.00076939798697543719269540758, 4.59061021334111790821842944484, 5.19966220173338409547411413810, 5.84104667582895084670343342048, 6.83292483927636345404920072429, 7.62275626876507036051073751582, 8.106055174109245415377310886242
