| L(s) = 1 | + (1.41 − 0.0725i)2-s + (−1.51 − 0.839i)3-s + (1.98 − 0.204i)4-s + (−1.38 − 1.57i)5-s + (−2.20 − 1.07i)6-s + (2.49 + 0.328i)7-s + (2.79 − 0.433i)8-s + (1.59 + 2.54i)9-s + (−2.06 − 2.12i)10-s + (−1.07 − 2.18i)11-s + (−3.18 − 1.35i)12-s + (−1.11 − 3.28i)13-s + (3.54 + 0.282i)14-s + (0.770 + 3.54i)15-s + (3.91 − 0.815i)16-s + (−1.22 − 1.22i)17-s + ⋯ |
| L(s) = 1 | + (0.998 − 0.0513i)2-s + (−0.874 − 0.484i)3-s + (0.994 − 0.102i)4-s + (−0.617 − 0.703i)5-s + (−0.898 − 0.439i)6-s + (0.941 + 0.124i)7-s + (0.988 − 0.153i)8-s + (0.530 + 0.847i)9-s + (−0.652 − 0.671i)10-s + (−0.324 − 0.658i)11-s + (−0.919 − 0.392i)12-s + (−0.309 − 0.912i)13-s + (0.947 + 0.0755i)14-s + (0.198 + 0.914i)15-s + (0.978 − 0.203i)16-s + (−0.297 − 0.297i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0706 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0706 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.44685 - 1.34804i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.44685 - 1.34804i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.41 + 0.0725i)T \) |
| 3 | \( 1 + (1.51 + 0.839i)T \) |
| good | 5 | \( 1 + (1.38 + 1.57i)T + (-0.652 + 4.95i)T^{2} \) |
| 7 | \( 1 + (-2.49 - 0.328i)T + (6.76 + 1.81i)T^{2} \) |
| 11 | \( 1 + (1.07 + 2.18i)T + (-6.69 + 8.72i)T^{2} \) |
| 13 | \( 1 + (1.11 + 3.28i)T + (-10.3 + 7.91i)T^{2} \) |
| 17 | \( 1 + (1.22 + 1.22i)T + 17iT^{2} \) |
| 19 | \( 1 + (-0.214 + 1.07i)T + (-17.5 - 7.27i)T^{2} \) |
| 23 | \( 1 + (0.00305 + 0.0232i)T + (-22.2 + 5.95i)T^{2} \) |
| 29 | \( 1 + (7.22 - 0.473i)T + (28.7 - 3.78i)T^{2} \) |
| 31 | \( 1 + (-7.56 + 4.36i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.63 - 8.22i)T + (-34.1 + 14.1i)T^{2} \) |
| 41 | \( 1 + (-0.138 - 1.05i)T + (-39.6 + 10.6i)T^{2} \) |
| 43 | \( 1 + (-1.20 + 0.596i)T + (26.1 - 34.1i)T^{2} \) |
| 47 | \( 1 + (-1.62 + 0.434i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (1.30 - 1.95i)T + (-20.2 - 48.9i)T^{2} \) |
| 59 | \( 1 + (-2.72 - 3.10i)T + (-7.70 + 58.4i)T^{2} \) |
| 61 | \( 1 + (0.179 + 2.73i)T + (-60.4 + 7.96i)T^{2} \) |
| 67 | \( 1 + (-13.8 - 6.82i)T + (40.7 + 53.1i)T^{2} \) |
| 71 | \( 1 + (-3.18 - 7.69i)T + (-50.2 + 50.2i)T^{2} \) |
| 73 | \( 1 + (5.78 - 13.9i)T + (-51.6 - 51.6i)T^{2} \) |
| 79 | \( 1 + (-2.42 - 9.04i)T + (-68.4 + 39.5i)T^{2} \) |
| 83 | \( 1 + (-6.39 - 5.60i)T + (10.8 + 82.2i)T^{2} \) |
| 89 | \( 1 + (-11.5 + 4.79i)T + (62.9 - 62.9i)T^{2} \) |
| 97 | \( 1 + (-1.38 - 0.802i)T + (48.5 + 84.0i)T^{2} \) |
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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
−11.04027605730352748548400641356, −10.00025522144043543492424418081, −8.234247322723618397578988917257, −7.84323389264381617913872759918, −6.72435233437872867050237269719, −5.59963508023026050510928931454, −5.03106816638608299474114611289, −4.16613675530321378411795008426, −2.53457302570458578308134165258, −0.960064782983126362465183316986,
1.95047097630436314861705948764, 3.59052675263793859745606164299, 4.45199203021415756255450083258, 5.15064989398410420563201951551, 6.31093324194928572707105442815, 7.14567209058830520024973710228, 7.84642482007605524499670240413, 9.418117007314417459384800452365, 10.61589661005638999853651697902, 11.00366892745929418420741838545
