| L(s) = 1 | + (0.540 − 0.841i)2-s + (1.37 + 1.05i)3-s + (−0.415 − 0.909i)4-s + (−0.959 − 0.281i)5-s + (1.63 − 0.583i)6-s + (2.43 + 2.10i)7-s + (−0.989 − 0.142i)8-s + (0.767 + 2.90i)9-s + (−0.755 + 0.654i)10-s + (−1.55 + 1.00i)11-s + (0.390 − 1.68i)12-s + (2.29 + 2.64i)13-s + (3.08 − 0.906i)14-s + (−1.01 − 1.40i)15-s + (−0.654 + 0.755i)16-s + (−1.24 + 2.72i)17-s + ⋯ |
| L(s) = 1 | + (0.382 − 0.594i)2-s + (0.792 + 0.609i)3-s + (−0.207 − 0.454i)4-s + (−0.429 − 0.125i)5-s + (0.665 − 0.238i)6-s + (0.919 + 0.796i)7-s + (−0.349 − 0.0503i)8-s + (0.255 + 0.966i)9-s + (−0.238 + 0.207i)10-s + (−0.469 + 0.301i)11-s + (0.112 − 0.487i)12-s + (0.635 + 0.733i)13-s + (0.825 − 0.242i)14-s + (−0.263 − 0.361i)15-s + (−0.163 + 0.188i)16-s + (−0.301 + 0.660i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 - 0.263i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.964 - 0.263i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.33493 + 0.313716i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.33493 + 0.313716i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.540 + 0.841i)T \) |
| 3 | \( 1 + (-1.37 - 1.05i)T \) |
| 5 | \( 1 + (0.959 + 0.281i)T \) |
| 23 | \( 1 + (1.94 + 4.38i)T \) |
| good | 7 | \( 1 + (-2.43 - 2.10i)T + (0.996 + 6.92i)T^{2} \) |
| 11 | \( 1 + (1.55 - 1.00i)T + (4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (-2.29 - 2.64i)T + (-1.85 + 12.8i)T^{2} \) |
| 17 | \( 1 + (1.24 - 2.72i)T + (-11.1 - 12.8i)T^{2} \) |
| 19 | \( 1 + (-5.86 + 2.68i)T + (12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (-2.52 - 1.15i)T + (18.9 + 21.9i)T^{2} \) |
| 31 | \( 1 + (-0.0402 + 0.280i)T + (-29.7 - 8.73i)T^{2} \) |
| 37 | \( 1 + (0.359 + 1.22i)T + (-31.1 + 20.0i)T^{2} \) |
| 41 | \( 1 + (-3.28 + 11.1i)T + (-34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (4.25 - 0.611i)T + (41.2 - 12.1i)T^{2} \) |
| 47 | \( 1 - 5.38iT - 47T^{2} \) |
| 53 | \( 1 + (-4.11 + 4.74i)T + (-7.54 - 52.4i)T^{2} \) |
| 59 | \( 1 + (-3.68 + 3.19i)T + (8.39 - 58.3i)T^{2} \) |
| 61 | \( 1 + (8.76 + 1.25i)T + (58.5 + 17.1i)T^{2} \) |
| 67 | \( 1 + (3.59 - 5.58i)T + (-27.8 - 60.9i)T^{2} \) |
| 71 | \( 1 + (2.11 - 3.28i)T + (-29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (-2.20 - 4.82i)T + (-47.8 + 55.1i)T^{2} \) |
| 79 | \( 1 + (-4.05 + 3.51i)T + (11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (11.1 - 3.28i)T + (69.8 - 44.8i)T^{2} \) |
| 89 | \( 1 + (-0.253 - 1.76i)T + (-85.3 + 25.0i)T^{2} \) |
| 97 | \( 1 + (-2.61 + 8.89i)T + (-81.6 - 52.4i)T^{2} \) |
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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
−10.63119081069270060021764642965, −9.641449244348324655165447354674, −8.748608188858488892802226937570, −8.289201088516855418711688323274, −7.14047791668411200193962351402, −5.62294815192983423484033360513, −4.75736610715912549315592975444, −4.00189395594770263014663633980, −2.80415983213966096419564316602, −1.78523431336258709114414491692,
1.17277563250191458234554786248, 2.95695646435497542769012153671, 3.80590788718097309615883979332, 4.95243089838413277788970829090, 6.05912500049330218438876006907, 7.20930322187177368763377228702, 7.82694232304042918943061671211, 8.205915856299395944503738866306, 9.355327932776100261166123290800, 10.42470577099747007984092113975
