| L(s) = 1 | + (1.5 + 0.866i)3-s + (1.5 + 2.59i)9-s + 3·11-s + 2·13-s − 5.19i·17-s − 5.19i·19-s + 6·23-s + 5.19i·27-s + 10.3i·29-s + 3.46i·31-s + (4.5 + 2.59i)33-s − 8·37-s + (3 + 1.73i)39-s + 5.19i·41-s − 3.46i·43-s + ⋯ |
| L(s) = 1 | + (0.866 + 0.499i)3-s + (0.5 + 0.866i)9-s + 0.904·11-s + 0.554·13-s − 1.26i·17-s − 1.19i·19-s + 1.25·23-s + 0.999i·27-s + 1.92i·29-s + 0.622i·31-s + (0.783 + 0.452i)33-s − 1.31·37-s + (0.480 + 0.277i)39-s + 0.811i·41-s − 0.528i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 - 0.5i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.866 - 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.448963120\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.448963120\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.5 - 0.866i)T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 - 3T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + 5.19iT - 17T^{2} \) |
| 19 | \( 1 + 5.19iT - 19T^{2} \) |
| 23 | \( 1 - 6T + 23T^{2} \) |
| 29 | \( 1 - 10.3iT - 29T^{2} \) |
| 31 | \( 1 - 3.46iT - 31T^{2} \) |
| 37 | \( 1 + 8T + 37T^{2} \) |
| 41 | \( 1 - 5.19iT - 41T^{2} \) |
| 43 | \( 1 + 3.46iT - 43T^{2} \) |
| 47 | \( 1 + 6T + 47T^{2} \) |
| 53 | \( 1 - 10.3iT - 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 - 8T + 61T^{2} \) |
| 67 | \( 1 + 12.1iT - 67T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 + T + 73T^{2} \) |
| 79 | \( 1 + 6.92iT - 79T^{2} \) |
| 83 | \( 1 + 9T + 83T^{2} \) |
| 89 | \( 1 - 5.19iT - 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−9.567972938619317561013746762792, −8.946556696855410891653361674147, −8.561096986885643761215032663331, −7.13625477715377596711215652509, −6.90407213384807497456515070758, −5.30339302420644617768063060101, −4.64677931818851887307434951068, −3.50249802131170676754177676213, −2.79897978593093859497624378701, −1.33592265847091080086535612718,
1.21580348837780398920690467909, 2.23714710739069456453926632505, 3.62616091572041836974630010664, 4.05808383313783800148740209705, 5.66419977881165032911271124808, 6.46396752624243489975258546626, 7.22448923444748345220654041407, 8.302801054496911758025187629576, 8.589486773596952376594479886760, 9.636056766947829873974712111778
