| L(s) = 1 | + (1.11 + 0.864i)2-s + (−0.0794 − 0.705i)3-s + (0.505 + 1.93i)4-s + (−1.39 − 0.670i)5-s + (0.520 − 0.858i)6-s + (3.80 + 3.03i)7-s + (−1.10 + 2.60i)8-s + (2.43 − 0.555i)9-s + (−0.978 − 1.95i)10-s + (−2.14 − 3.41i)11-s + (1.32 − 0.510i)12-s + (−0.467 + 2.05i)13-s + (1.63 + 6.68i)14-s + (−0.362 + 1.03i)15-s + (−3.48 + 1.95i)16-s + (−1.70 + 1.70i)17-s + ⋯ |
| L(s) = 1 | + (0.791 + 0.611i)2-s + (−0.0458 − 0.407i)3-s + (0.252 + 0.967i)4-s + (−0.622 − 0.299i)5-s + (0.212 − 0.350i)6-s + (1.43 + 1.14i)7-s + (−0.391 + 0.920i)8-s + (0.811 − 0.185i)9-s + (−0.309 − 0.617i)10-s + (−0.646 − 1.02i)11-s + (0.382 − 0.147i)12-s + (−0.129 + 0.568i)13-s + (0.437 + 1.78i)14-s + (−0.0934 + 0.267i)15-s + (−0.872 + 0.489i)16-s + (−0.414 + 0.414i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.643 - 0.765i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.643 - 0.765i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.68417 + 0.784985i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.68417 + 0.784985i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.11 - 0.864i)T \) |
| 29 | \( 1 + (3.52 - 4.07i)T \) |
| good | 3 | \( 1 + (0.0794 + 0.705i)T + (-2.92 + 0.667i)T^{2} \) |
| 5 | \( 1 + (1.39 + 0.670i)T + (3.11 + 3.90i)T^{2} \) |
| 7 | \( 1 + (-3.80 - 3.03i)T + (1.55 + 6.82i)T^{2} \) |
| 11 | \( 1 + (2.14 + 3.41i)T + (-4.77 + 9.91i)T^{2} \) |
| 13 | \( 1 + (0.467 - 2.05i)T + (-11.7 - 5.64i)T^{2} \) |
| 17 | \( 1 + (1.70 - 1.70i)T - 17iT^{2} \) |
| 19 | \( 1 + (-5.17 - 0.582i)T + (18.5 + 4.22i)T^{2} \) |
| 23 | \( 1 + (3.77 + 7.83i)T + (-14.3 + 17.9i)T^{2} \) |
| 31 | \( 1 + (1.07 + 3.07i)T + (-24.2 + 19.3i)T^{2} \) |
| 37 | \( 1 + (3.69 + 2.31i)T + (16.0 + 33.3i)T^{2} \) |
| 41 | \( 1 + (-0.300 - 0.300i)T + 41iT^{2} \) |
| 43 | \( 1 + (-2.37 + 6.79i)T + (-33.6 - 26.8i)T^{2} \) |
| 47 | \( 1 + (0.736 + 1.17i)T + (-20.3 + 42.3i)T^{2} \) |
| 53 | \( 1 + (1.55 - 3.21i)T + (-33.0 - 41.4i)T^{2} \) |
| 59 | \( 1 - 10.7T + 59T^{2} \) |
| 61 | \( 1 + (-1.62 + 0.183i)T + (59.4 - 13.5i)T^{2} \) |
| 67 | \( 1 + (11.0 - 2.52i)T + (60.3 - 29.0i)T^{2} \) |
| 71 | \( 1 + (1.38 - 6.07i)T + (-63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (9.61 + 3.36i)T + (57.0 + 45.5i)T^{2} \) |
| 79 | \( 1 + (-3.84 + 6.12i)T + (-34.2 - 71.1i)T^{2} \) |
| 83 | \( 1 + (-9.71 - 12.1i)T + (-18.4 + 80.9i)T^{2} \) |
| 89 | \( 1 + (-7.35 + 2.57i)T + (69.5 - 55.4i)T^{2} \) |
| 97 | \( 1 + (1.84 - 16.3i)T + (-94.5 - 21.5i)T^{2} \) |
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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
−12.17560763392379504951363184714, −11.86867789115815216199291281792, −10.80923482238728102224750693927, −8.858099794833534994199409904599, −8.187588261335355852985752712220, −7.39811230106667275497025099023, −6.03888289770321929938139061209, −5.07462319494247893059139771464, −4.02568188586052103698944959766, −2.21278295453544156025338288979,
1.66258908755369160708637948097, 3.59640127062677207812574324931, 4.55452538660901754592363137601, 5.27119532230594778077180937513, 7.42056424692267394824373216805, 7.52885926129796723635432884600, 9.688748778377258087888785498445, 10.33072415169835313084799532188, 11.23869101973556972493767946718, 11.79907884248629727485104236388
