| L(s) = 1 | + (−0.984 + 0.173i)2-s + (−1.56 − 0.743i)3-s + (0.939 − 0.342i)4-s + (−0.941 − 0.789i)5-s + (1.66 + 0.460i)6-s + (2.62 + 0.339i)7-s + (−0.866 + 0.5i)8-s + (1.89 + 2.32i)9-s + (1.06 + 0.614i)10-s + (−1.50 − 1.79i)11-s + (−1.72 − 0.163i)12-s + (0.429 + 0.0757i)13-s + (−2.64 + 0.121i)14-s + (0.885 + 1.93i)15-s + (0.766 − 0.642i)16-s + (1.74 − 3.01i)17-s + ⋯ |
| L(s) = 1 | + (−0.696 + 0.122i)2-s + (−0.903 − 0.429i)3-s + (0.469 − 0.171i)4-s + (−0.420 − 0.353i)5-s + (0.681 + 0.187i)6-s + (0.991 + 0.128i)7-s + (−0.306 + 0.176i)8-s + (0.631 + 0.775i)9-s + (0.336 + 0.194i)10-s + (−0.454 − 0.541i)11-s + (−0.497 − 0.0471i)12-s + (0.119 + 0.0210i)13-s + (−0.706 + 0.0324i)14-s + (0.228 + 0.499i)15-s + (0.191 − 0.160i)16-s + (0.422 − 0.732i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.178 + 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.178 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.376985 - 0.451580i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.376985 - 0.451580i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.984 - 0.173i)T \) |
| 3 | \( 1 + (1.56 + 0.743i)T \) |
| 7 | \( 1 + (-2.62 - 0.339i)T \) |
| good | 5 | \( 1 + (0.941 + 0.789i)T + (0.868 + 4.92i)T^{2} \) |
| 11 | \( 1 + (1.50 + 1.79i)T + (-1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-0.429 - 0.0757i)T + (12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (-1.74 + 3.01i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.97 - 1.13i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.85 + 5.08i)T + (-17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (2.08 - 0.367i)T + (27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (2.87 + 7.89i)T + (-23.7 + 19.9i)T^{2} \) |
| 37 | \( 1 + (-2.71 + 4.69i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.273 + 1.55i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (1.56 - 1.31i)T + (7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (-4.45 - 1.62i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + 0.117iT - 53T^{2} \) |
| 59 | \( 1 + (9.22 + 7.74i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (3.82 - 10.5i)T + (-46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (-1.46 + 8.32i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (1.77 + 1.02i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-6.00 + 3.46i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (0.916 + 5.19i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (-1.50 - 8.54i)T + (-77.9 + 28.3i)T^{2} \) |
| 89 | \( 1 + (-7.35 - 12.7i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-9.82 - 11.7i)T + (-16.8 + 95.5i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.05568814128847066124388678584, −10.47112822291824300120013056147, −9.177469045116060536126677032368, −8.023537853363314854851553351945, −7.67405950073319667199326126140, −6.32628340890515669425703787708, −5.42531202511598142482262000634, −4.34035159088106957445651656261, −2.18077190640041740169724925776, −0.57593813769841326583886426401,
1.58357071537984339236250668923, 3.56174022261516972464318797064, 4.77895140471086885808219158164, 5.84159019682646034792174715784, 7.10772478772537058057313778670, 7.80856966888998096419602104019, 8.940923297240468518753268240188, 10.06032874714716211003044219141, 10.72063592230871970470838278634, 11.38884718202196110107430480775
