| L(s) = 1 | + 1.41i·3-s − 2.82i·7-s + 0.999·9-s − 0.828·11-s − 3.41i·13-s + 4.82i·17-s + 19-s + 4.00·21-s − 4i·23-s + 5.65i·27-s + 0.828·29-s − 1.17i·33-s + 10.2i·37-s + 4.82·39-s − 0.828·41-s + ⋯ |
| L(s) = 1 | + 0.816i·3-s − 1.06i·7-s + 0.333·9-s − 0.249·11-s − 0.946i·13-s + 1.17i·17-s + 0.229·19-s + 0.872·21-s − 0.834i·23-s + 1.08i·27-s + 0.153·29-s − 0.203i·33-s + 1.68i·37-s + 0.773·39-s − 0.129·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.801422817\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.801422817\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 1.41iT - 3T^{2} \) |
| 7 | \( 1 + 2.82iT - 7T^{2} \) |
| 11 | \( 1 + 0.828T + 11T^{2} \) |
| 13 | \( 1 + 3.41iT - 13T^{2} \) |
| 17 | \( 1 - 4.82iT - 17T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 - 0.828T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 10.2iT - 37T^{2} \) |
| 41 | \( 1 + 0.828T + 41T^{2} \) |
| 43 | \( 1 + 2.82iT - 43T^{2} \) |
| 47 | \( 1 + 8.48iT - 47T^{2} \) |
| 53 | \( 1 + 13.0iT - 53T^{2} \) |
| 59 | \( 1 + 2.82T + 59T^{2} \) |
| 61 | \( 1 + 1.65T + 61T^{2} \) |
| 67 | \( 1 + 9.41iT - 67T^{2} \) |
| 71 | \( 1 - 15.3T + 71T^{2} \) |
| 73 | \( 1 + 12.1iT - 73T^{2} \) |
| 79 | \( 1 - 9.17T + 79T^{2} \) |
| 83 | \( 1 + 8iT - 83T^{2} \) |
| 89 | \( 1 - 3.17T + 89T^{2} \) |
| 97 | \( 1 + 2.24iT - 97T^{2} \) |
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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
−8.326272679492526259235753638618, −7.86224691257874403037803857021, −6.93720308680941416236691715802, −6.29431784121526327737193564038, −5.19227500985419386386335802657, −4.65645472571003325052207961880, −3.76269807779508491356208541248, −3.27316283202735744063104701357, −1.85698130575178927096587797715, −0.61189500846417655766494429161,
1.04791710551625612844086626317, 2.11406884940090672202995301269, 2.73278542218766200761530548913, 3.95494001542263779348122537322, 4.88077886279564313732668701000, 5.67182088617579914098675420952, 6.37350114615724197602382549379, 7.22003583119976322323211506905, 7.60952660003720982111160762874, 8.535322274329208126491057695366
