L(s) = 1 | + (0.309 + 0.951i)2-s + (1.48 + 0.898i)3-s + (−0.809 + 0.587i)4-s + (3.37 + 1.09i)5-s + (−0.396 + 1.68i)6-s + (0.587 + 0.809i)7-s + (−0.809 − 0.587i)8-s + (1.38 + 2.66i)9-s + 3.54i·10-s + (−1.81 − 2.77i)11-s + (−1.72 + 0.143i)12-s + (−5.58 + 1.81i)13-s + (−0.587 + 0.809i)14-s + (4.01 + 4.65i)15-s + (0.309 − 0.951i)16-s + (2.31 − 7.12i)17-s + ⋯ |
L(s) = 1 | + (0.218 + 0.672i)2-s + (0.854 + 0.518i)3-s + (−0.404 + 0.293i)4-s + (1.50 + 0.490i)5-s + (−0.161 + 0.688i)6-s + (0.222 + 0.305i)7-s + (−0.286 − 0.207i)8-s + (0.462 + 0.886i)9-s + 1.12i·10-s + (−0.546 − 0.837i)11-s + (−0.498 + 0.0414i)12-s + (−1.54 + 0.503i)13-s + (−0.157 + 0.216i)14-s + (1.03 + 1.20i)15-s + (0.0772 − 0.237i)16-s + (0.561 − 1.72i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0547 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0547 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.63089 + 1.72269i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.63089 + 1.72269i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.309 - 0.951i)T \) |
| 3 | \( 1 + (-1.48 - 0.898i)T \) |
| 7 | \( 1 + (-0.587 - 0.809i)T \) |
| 11 | \( 1 + (1.81 + 2.77i)T \) |
good | 5 | \( 1 + (-3.37 - 1.09i)T + (4.04 + 2.93i)T^{2} \) |
| 13 | \( 1 + (5.58 - 1.81i)T + (10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (-2.31 + 7.12i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (-2.87 + 3.95i)T + (-5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + 3.38iT - 23T^{2} \) |
| 29 | \( 1 + (4.51 - 3.27i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (0.123 + 0.381i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (8.42 - 6.11i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (0.577 + 0.419i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 0.722iT - 43T^{2} \) |
| 47 | \( 1 + (-0.656 + 0.903i)T + (-14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (0.334 - 0.108i)T + (42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (-0.843 - 1.16i)T + (-18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (1.21 + 0.393i)T + (49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 - 9.40T + 67T^{2} \) |
| 71 | \( 1 + (-3.46 - 1.12i)T + (57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (1.72 + 2.38i)T + (-22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (-11.2 + 3.65i)T + (63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (3.11 - 9.57i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 - 4.97iT - 89T^{2} \) |
| 97 | \( 1 + (0.155 + 0.479i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−11.08913347883294459705741395063, −9.968353716945920375577684154277, −9.496968636966356315070505138137, −8.734942873872131831570382259264, −7.49505180955795117965257350498, −6.78648205991033481465769428383, −5.27465670029317893100869892323, −5.03222973523932779690301379078, −3.08728281236138897054672294903, −2.36101477055815314402421461533,
1.60039845886661667579119971485, 2.25213164400008195824469119676, 3.67503575256767997420267876302, 5.09101739626569858053168937526, 5.88868542921729196894708079103, 7.33065212714555601836366533163, 8.133204609097727716354417728492, 9.348469434688128608710026538244, 9.921099921752706937119926708659, 10.42146454022669015306764791552