| L(s) = 1 | + (−0.572 + 1.29i)2-s + (1.73 − 0.0801i)3-s + (−1.34 − 1.48i)4-s + (3.44 − 1.60i)5-s + (−0.887 + 2.28i)6-s + (−0.0228 + 0.129i)7-s + (2.68 − 0.890i)8-s + (2.98 − 0.277i)9-s + (0.104 + 5.37i)10-s + (−4.46 − 2.08i)11-s + (−2.44 − 2.45i)12-s + (0.237 − 2.71i)13-s + (−0.154 − 0.103i)14-s + (5.83 − 3.05i)15-s + (−0.386 + 3.98i)16-s + (1.14 − 0.663i)17-s + ⋯ |
| L(s) = 1 | + (−0.404 + 0.914i)2-s + (0.998 − 0.0462i)3-s + (−0.672 − 0.740i)4-s + (1.54 − 0.719i)5-s + (−0.362 + 0.932i)6-s + (−0.00863 + 0.0489i)7-s + (0.949 − 0.314i)8-s + (0.995 − 0.0924i)9-s + (0.0330 + 1.70i)10-s + (−1.34 − 0.628i)11-s + (−0.705 − 0.708i)12-s + (0.0659 − 0.753i)13-s + (−0.0412 − 0.0277i)14-s + (1.50 − 0.789i)15-s + (−0.0966 + 0.995i)16-s + (0.278 − 0.160i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.958 - 0.285i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.958 - 0.285i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.78463 + 0.260009i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.78463 + 0.260009i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.572 - 1.29i)T \) |
| 3 | \( 1 + (-1.73 + 0.0801i)T \) |
| good | 5 | \( 1 + (-3.44 + 1.60i)T + (3.21 - 3.83i)T^{2} \) |
| 7 | \( 1 + (0.0228 - 0.129i)T + (-6.57 - 2.39i)T^{2} \) |
| 11 | \( 1 + (4.46 + 2.08i)T + (7.07 + 8.42i)T^{2} \) |
| 13 | \( 1 + (-0.237 + 2.71i)T + (-12.8 - 2.25i)T^{2} \) |
| 17 | \( 1 + (-1.14 + 0.663i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.48 - 0.397i)T + (16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (5.04 - 0.888i)T + (21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-0.743 - 8.49i)T + (-28.5 + 5.03i)T^{2} \) |
| 31 | \( 1 + (-5.22 + 0.921i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (1.59 - 5.95i)T + (-32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (3.95 - 3.31i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (2.27 - 4.88i)T + (-27.6 - 32.9i)T^{2} \) |
| 47 | \( 1 + (1.24 - 7.05i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (6.18 - 6.18i)T - 53iT^{2} \) |
| 59 | \( 1 + (-1.95 - 4.20i)T + (-37.9 + 45.1i)T^{2} \) |
| 61 | \( 1 + (3.15 + 4.50i)T + (-20.8 + 57.3i)T^{2} \) |
| 67 | \( 1 + (-13.0 - 1.14i)T + (65.9 + 11.6i)T^{2} \) |
| 71 | \( 1 + (2.06 - 1.19i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (1.73 + 1.00i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.04 + 1.24i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (0.262 - 0.0229i)T + (81.7 - 14.4i)T^{2} \) |
| 89 | \( 1 + (-4.50 + 7.81i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-17.6 + 6.43i)T + (74.3 - 62.3i)T^{2} \) |
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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
−10.51353390265502158549338865803, −10.05811595043405678173773091678, −9.240757173167529589827404566808, −8.373552222746451647286157244574, −7.86510193773476292640514976049, −6.46895521486509806686175033509, −5.57249642645293496948073703751, −4.75648763877599463282984972261, −2.86697489643326681095337815922, −1.41030189264884979052743637177,
2.09347694105997902177233635882, 2.38284045614920267677866721995, 3.81662067920502643711962118493, 5.13797255782109071854342452216, 6.61384558003828703297892949250, 7.69262872927830687349425125844, 8.604308903512494030224482581165, 9.655756246131376125871711500491, 10.09513110091403202527178150569, 10.63008455046743928412821611988
