| L(s) = 1 | + (0.654 − 0.755i)2-s + (1.13 + 1.76i)3-s + (−0.142 − 0.989i)4-s + (1.26 + 2.76i)5-s + (2.07 + 0.298i)6-s + (−2.15 + 1.53i)7-s + (−0.841 − 0.540i)8-s + (−0.580 + 1.27i)9-s + (2.91 + 0.856i)10-s + (−1.26 + 1.09i)11-s + (1.58 − 1.37i)12-s + (0.771 − 2.62i)13-s + (−0.254 + 2.63i)14-s + (−3.44 + 5.36i)15-s + (−0.959 + 0.281i)16-s + (0.410 − 2.85i)17-s + ⋯ |
| L(s) = 1 | + (0.463 − 0.534i)2-s + (0.654 + 1.01i)3-s + (−0.0711 − 0.494i)4-s + (0.564 + 1.23i)5-s + (0.847 + 0.121i)6-s + (−0.815 + 0.578i)7-s + (−0.297 − 0.191i)8-s + (−0.193 + 0.423i)9-s + (0.922 + 0.270i)10-s + (−0.380 + 0.329i)11-s + (0.457 − 0.396i)12-s + (0.214 − 0.729i)13-s + (−0.0681 + 0.703i)14-s + (−0.889 + 1.38i)15-s + (−0.239 + 0.0704i)16-s + (0.0994 − 0.691i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.692 - 0.721i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.692 - 0.721i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.83531 + 0.782221i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.83531 + 0.782221i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.654 + 0.755i)T \) |
| 7 | \( 1 + (2.15 - 1.53i)T \) |
| 23 | \( 1 + (-2.87 + 3.83i)T \) |
| good | 3 | \( 1 + (-1.13 - 1.76i)T + (-1.24 + 2.72i)T^{2} \) |
| 5 | \( 1 + (-1.26 - 2.76i)T + (-3.27 + 3.77i)T^{2} \) |
| 11 | \( 1 + (1.26 - 1.09i)T + (1.56 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-0.771 + 2.62i)T + (-10.9 - 7.02i)T^{2} \) |
| 17 | \( 1 + (-0.410 + 2.85i)T + (-16.3 - 4.78i)T^{2} \) |
| 19 | \( 1 + (-0.541 - 3.76i)T + (-18.2 + 5.35i)T^{2} \) |
| 29 | \( 1 + (-1.00 + 6.98i)T + (-27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (-2.29 + 3.57i)T + (-12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + (-9.45 - 4.31i)T + (24.2 + 27.9i)T^{2} \) |
| 41 | \( 1 + (5.40 - 2.46i)T + (26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (-0.167 - 0.260i)T + (-17.8 + 39.1i)T^{2} \) |
| 47 | \( 1 - 7.01iT - 47T^{2} \) |
| 53 | \( 1 + (1.40 + 4.77i)T + (-44.5 + 28.6i)T^{2} \) |
| 59 | \( 1 + (-1.21 + 4.14i)T + (-49.6 - 31.8i)T^{2} \) |
| 61 | \( 1 + (2.82 + 1.81i)T + (25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (11.4 + 9.93i)T + (9.53 + 66.3i)T^{2} \) |
| 71 | \( 1 + (9.46 - 10.9i)T + (-10.1 - 70.2i)T^{2} \) |
| 73 | \( 1 + (8.14 - 1.17i)T + (70.0 - 20.5i)T^{2} \) |
| 79 | \( 1 + (-0.420 + 1.43i)T + (-66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (7.17 - 15.7i)T + (-54.3 - 62.7i)T^{2} \) |
| 89 | \( 1 + (1.65 - 1.06i)T + (36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + (3.61 + 7.91i)T + (-63.5 + 73.3i)T^{2} \) |
| 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
−11.62024035584171714377039009759, −10.54484527052572947867302044490, −9.921825972657691001144656825362, −9.498287964265658211471640408425, −8.127015666257480510335762423877, −6.61979755809301770329973346388, −5.77044165935127588728132376864, −4.39178157916534952414967703348, −3.05396810059770536871426436862, −2.69552605182996775994468042758,
1.37326286450966726058863681146, 3.02852349489232844565511150392, 4.50111202840303101879000738968, 5.67223842397345188487180504573, 6.76570045093864365115674426694, 7.52189542635948011460311866097, 8.693144988082606043931880562614, 9.143405536456142898407809298388, 10.52556954721855160305260972858, 11.98321122138219977538577359207
