| L(s) = 1 | + (−0.337 + 0.774i)2-s + (0.874 + 0.942i)4-s + (3.34 + 0.504i)5-s + (−1.95 − 1.81i)7-s + (−2.62 + 0.917i)8-s + (−1.52 + 2.42i)10-s + (4.07 − 3.51i)11-s + (−2.27 + 3.33i)13-s + (2.06 − 0.900i)14-s + (−0.0167 + 0.224i)16-s + (4.02 + 4.02i)17-s + (0.0254 + 0.0160i)19-s + (2.45 + 3.59i)20-s + (1.34 + 4.34i)22-s + (6.17 + 2.42i)23-s + ⋯ |
| L(s) = 1 | + (−0.238 + 0.547i)2-s + (0.437 + 0.471i)4-s + (1.49 + 0.225i)5-s + (−0.738 − 0.685i)7-s + (−0.926 + 0.324i)8-s + (−0.481 + 0.766i)10-s + (1.22 − 1.05i)11-s + (−0.631 + 0.926i)13-s + (0.551 − 0.240i)14-s + (−0.00419 + 0.0560i)16-s + (0.975 + 0.975i)17-s + (0.00584 + 0.00367i)19-s + (0.548 + 0.804i)20-s + (0.285 + 0.926i)22-s + (1.28 + 0.504i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 783 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.312 - 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 783 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.312 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.53186 + 1.10908i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.53186 + 1.10908i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 29 | \( 1 + (5.34 - 0.678i)T \) |
| good | 2 | \( 1 + (0.337 - 0.774i)T + (-1.36 - 1.46i)T^{2} \) |
| 5 | \( 1 + (-3.34 - 0.504i)T + (4.77 + 1.47i)T^{2} \) |
| 7 | \( 1 + (1.95 + 1.81i)T + (0.523 + 6.98i)T^{2} \) |
| 11 | \( 1 + (-4.07 + 3.51i)T + (1.63 - 10.8i)T^{2} \) |
| 13 | \( 1 + (2.27 - 3.33i)T + (-4.74 - 12.1i)T^{2} \) |
| 17 | \( 1 + (-4.02 - 4.02i)T + 17iT^{2} \) |
| 19 | \( 1 + (-0.0254 - 0.0160i)T + (8.24 + 17.1i)T^{2} \) |
| 23 | \( 1 + (-6.17 - 2.42i)T + (16.8 + 15.6i)T^{2} \) |
| 31 | \( 1 + (-2.32 + 1.71i)T + (9.13 - 29.6i)T^{2} \) |
| 37 | \( 1 + (0.331 + 0.947i)T + (-28.9 + 23.0i)T^{2} \) |
| 41 | \( 1 + (-1.80 - 0.483i)T + (35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (-0.869 + 1.17i)T + (-12.6 - 41.0i)T^{2} \) |
| 47 | \( 1 + (-4.17 - 4.85i)T + (-7.00 + 46.4i)T^{2} \) |
| 53 | \( 1 + (3.94 + 3.14i)T + (11.7 + 51.6i)T^{2} \) |
| 59 | \( 1 + (3.11 - 1.79i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (0.0471 - 0.00176i)T + (60.8 - 4.55i)T^{2} \) |
| 67 | \( 1 + (15.6 - 1.16i)T + (66.2 - 9.98i)T^{2} \) |
| 71 | \( 1 + (-4.81 + 2.31i)T + (44.2 - 55.5i)T^{2} \) |
| 73 | \( 1 + (0.355 - 3.15i)T + (-71.1 - 16.2i)T^{2} \) |
| 79 | \( 1 + (-0.115 + 0.612i)T + (-73.5 - 28.8i)T^{2} \) |
| 83 | \( 1 + (-2.29 + 7.43i)T + (-68.5 - 46.7i)T^{2} \) |
| 89 | \( 1 + (-9.35 + 1.05i)T + (86.7 - 19.8i)T^{2} \) |
| 97 | \( 1 + (-7.40 - 14.0i)T + (-54.6 + 80.1i)T^{2} \) |
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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
−10.34265333529622974886242428890, −9.309357334044323418023029661979, −9.076556889528586214353270785906, −7.71274358857995416756647412034, −6.79933532748245447010441333424, −6.29253148833534341825559483197, −5.58350003634398553321918837810, −3.85540210392207568319460371873, −2.94416300562311553403092952460, −1.53900887751996281971756845373,
1.17348633449415521653239198862, 2.31388043688844436329033388751, 3.09680457490588258792203860744, 4.98571189582408423076333849079, 5.73863287601215032201370032175, 6.49160980660925016891045647916, 7.32307927119019279844319692276, 9.097649582054498137354217692885, 9.412954201619330325799921148861, 9.947045566205475788927168711771
