| L(s) = 1 | + (1.07 + 0.965i)2-s + (0.994 − 0.104i)3-s + (0.00859 + 0.0817i)4-s + (1.18 + 1.89i)5-s + (1.16 + 0.848i)6-s + (0.513 − 2.59i)7-s + (1.62 − 2.23i)8-s + (0.978 − 0.207i)9-s + (−0.566 + 3.17i)10-s + (3.57 + 0.758i)11-s + (0.0170 + 0.0803i)12-s + (−5.06 − 1.64i)13-s + (3.05 − 2.28i)14-s + (1.37 + 1.76i)15-s + (4.06 − 0.864i)16-s + (−1.82 − 4.10i)17-s + ⋯ |
| L(s) = 1 | + (0.758 + 0.682i)2-s + (0.574 − 0.0603i)3-s + (0.00429 + 0.0408i)4-s + (0.528 + 0.849i)5-s + (0.476 + 0.346i)6-s + (0.194 − 0.980i)7-s + (0.575 − 0.791i)8-s + (0.326 − 0.0693i)9-s + (−0.178 + 1.00i)10-s + (1.07 + 0.228i)11-s + (0.00493 + 0.0232i)12-s + (−1.40 − 0.456i)13-s + (0.816 − 0.611i)14-s + (0.354 + 0.455i)15-s + (1.01 − 0.216i)16-s + (−0.443 − 0.995i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.878 - 0.478i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.878 - 0.478i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.72234 + 0.693214i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.72234 + 0.693214i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.994 + 0.104i)T \) |
| 5 | \( 1 + (-1.18 - 1.89i)T \) |
| 7 | \( 1 + (-0.513 + 2.59i)T \) |
| good | 2 | \( 1 + (-1.07 - 0.965i)T + (0.209 + 1.98i)T^{2} \) |
| 11 | \( 1 + (-3.57 - 0.758i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (5.06 + 1.64i)T + (10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (1.82 + 4.10i)T + (-11.3 + 12.6i)T^{2} \) |
| 19 | \( 1 + (0.833 - 7.93i)T + (-18.5 - 3.95i)T^{2} \) |
| 23 | \( 1 + (-1.04 - 0.943i)T + (2.40 + 22.8i)T^{2} \) |
| 29 | \( 1 + (6.30 - 4.57i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-1.29 + 0.574i)T + (20.7 - 23.0i)T^{2} \) |
| 37 | \( 1 + (-1.14 - 5.37i)T + (-33.8 + 15.0i)T^{2} \) |
| 41 | \( 1 + (0.420 - 1.29i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + 4.76iT - 43T^{2} \) |
| 47 | \( 1 + (2.15 - 4.83i)T + (-31.4 - 34.9i)T^{2} \) |
| 53 | \( 1 + (-9.77 + 1.02i)T + (51.8 - 11.0i)T^{2} \) |
| 59 | \( 1 + (6.96 + 7.73i)T + (-6.16 + 58.6i)T^{2} \) |
| 61 | \( 1 + (2.46 - 2.73i)T + (-6.37 - 60.6i)T^{2} \) |
| 67 | \( 1 + (0.891 + 2.00i)T + (-44.8 + 49.7i)T^{2} \) |
| 71 | \( 1 + (0.0617 - 0.0448i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-2.16 + 10.1i)T + (-66.6 - 29.6i)T^{2} \) |
| 79 | \( 1 + (4.65 + 2.07i)T + (52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (2.95 - 4.06i)T + (-25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (6.80 - 7.55i)T + (-9.30 - 88.5i)T^{2} \) |
| 97 | \( 1 + (4.21 + 5.80i)T + (-29.9 + 92.2i)T^{2} \) |
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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
−10.71734667907174268821554662691, −9.959409827285135638070791737681, −9.424539984026543931910833153782, −7.75063868661848857842266514638, −7.16454739357427920329765155114, −6.49234947077623067362405900351, −5.33930884200872134234223461092, −4.27286014662637345009256795713, −3.30297379288914176881764861185, −1.67656722353264433993210840485,
1.90861105929484606788706275248, 2.62871221157350994738641120321, 4.14022913140168194960793527619, 4.78868568070883706829186127746, 5.85068919399420705607093004884, 7.18146701363373459628875460492, 8.475627838161062399226646988513, 8.992589368206451062778094224678, 9.741213333734541638959520515493, 11.12518787468954960620863544211
