| L(s) = 1 | + (−0.989 + 0.142i)2-s + (0.909 + 0.415i)3-s + (0.959 − 0.281i)4-s + (2.01 − 0.967i)5-s + (−0.959 − 0.281i)6-s + (−0.0999 − 0.155i)7-s + (−0.909 + 0.415i)8-s + (0.654 + 0.755i)9-s + (−1.85 + 1.24i)10-s + (−0.591 + 4.11i)11-s + (0.989 + 0.142i)12-s + (−3.08 + 4.80i)13-s + (0.121 + 0.139i)14-s + (2.23 − 0.0427i)15-s + (0.841 − 0.540i)16-s + (−1.78 + 6.08i)17-s + ⋯ |
| L(s) = 1 | + (−0.699 + 0.100i)2-s + (0.525 + 0.239i)3-s + (0.479 − 0.140i)4-s + (0.901 − 0.432i)5-s + (−0.391 − 0.115i)6-s + (−0.0377 − 0.0588i)7-s + (−0.321 + 0.146i)8-s + (0.218 + 0.251i)9-s + (−0.587 + 0.393i)10-s + (−0.178 + 1.24i)11-s + (0.285 + 0.0410i)12-s + (−0.855 + 1.33i)13-s + (0.0323 + 0.0373i)14-s + (0.577 − 0.0110i)15-s + (0.210 − 0.135i)16-s + (−0.433 + 1.47i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.543 - 0.839i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.543 - 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.26544 + 0.688018i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.26544 + 0.688018i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.989 - 0.142i)T \) |
| 3 | \( 1 + (-0.909 - 0.415i)T \) |
| 5 | \( 1 + (-2.01 + 0.967i)T \) |
| 23 | \( 1 + (-4.79 + 0.0673i)T \) |
| good | 7 | \( 1 + (0.0999 + 0.155i)T + (-2.90 + 6.36i)T^{2} \) |
| 11 | \( 1 + (0.591 - 4.11i)T + (-10.5 - 3.09i)T^{2} \) |
| 13 | \( 1 + (3.08 - 4.80i)T + (-5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (1.78 - 6.08i)T + (-14.3 - 9.19i)T^{2} \) |
| 19 | \( 1 + (-2.06 + 0.605i)T + (15.9 - 10.2i)T^{2} \) |
| 29 | \( 1 + (-4.70 - 1.38i)T + (24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (2.03 + 4.46i)T + (-20.3 + 23.4i)T^{2} \) |
| 37 | \( 1 + (-4.72 + 4.09i)T + (5.26 - 36.6i)T^{2} \) |
| 41 | \( 1 + (1.65 - 1.90i)T + (-5.83 - 40.5i)T^{2} \) |
| 43 | \( 1 + (-4.80 - 2.19i)T + (28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 - 2.44iT - 47T^{2} \) |
| 53 | \( 1 + (6.66 + 10.3i)T + (-22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (-9.13 - 5.87i)T + (24.5 + 53.6i)T^{2} \) |
| 61 | \( 1 + (5.02 + 11.0i)T + (-39.9 + 46.1i)T^{2} \) |
| 67 | \( 1 + (8.92 - 1.28i)T + (64.2 - 18.8i)T^{2} \) |
| 71 | \( 1 + (-0.970 - 6.74i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (4.11 + 13.9i)T + (-61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (-13.8 - 8.88i)T + (32.8 + 71.8i)T^{2} \) |
| 83 | \( 1 + (-0.426 + 0.369i)T + (11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (7.29 - 15.9i)T + (-58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + (1.35 + 1.17i)T + (13.8 + 96.0i)T^{2} \) |
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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
−10.25335687947365497682411284429, −9.562214062189042427120827187016, −9.144594637437736160431901337240, −8.168810064173239762117461280351, −7.15244348752711308109868959168, −6.41385130729246474099892601409, −5.09850235497362214746351248750, −4.22531972960596437833028880490, −2.47207984773918001065117544407, −1.67577084349385323215824837293,
0.940689661917581011858955113955, 2.74307858357844311256922592799, 2.98378793334570311558303219273, 5.06500309104413503147960388384, 5.98325310891132273713090532322, 7.04642254458136971315983703360, 7.72964451182986672930341123422, 8.783078249564293232788800419048, 9.386618559560611108231420270838, 10.26524713985859754189235863755
