| L(s) = 1 | + (0.453 + 0.891i)2-s + (3.07 − 0.739i)3-s + (−0.587 + 0.809i)4-s + (−0.951 − 0.0748i)5-s + (2.05 + 2.40i)6-s + (−0.0553 + 0.230i)7-s + (−0.987 − 0.156i)8-s + (6.26 − 3.18i)9-s + (−0.365 − 0.881i)10-s + (2.33 − 2.35i)11-s + (−1.21 + 2.92i)12-s + (−1.35 − 0.439i)13-s + (−0.230 + 0.0553i)14-s + (−2.98 + 0.472i)15-s + (−0.309 − 0.951i)16-s + (0.277 + 4.11i)17-s + ⋯ |
| L(s) = 1 | + (0.321 + 0.630i)2-s + (1.77 − 0.426i)3-s + (−0.293 + 0.404i)4-s + (−0.425 − 0.0334i)5-s + (0.839 + 0.982i)6-s + (−0.0209 + 0.0871i)7-s + (−0.349 − 0.0553i)8-s + (2.08 − 1.06i)9-s + (−0.115 − 0.278i)10-s + (0.704 − 0.709i)11-s + (−0.349 + 0.844i)12-s + (−0.375 − 0.122i)13-s + (−0.0616 + 0.0147i)14-s + (−0.770 + 0.122i)15-s + (−0.0772 − 0.237i)16-s + (0.0673 + 0.997i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 374 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.888 - 0.458i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 374 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.888 - 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.42231 + 0.588713i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.42231 + 0.588713i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.453 - 0.891i)T \) |
| 11 | \( 1 + (-2.33 + 2.35i)T \) |
| 17 | \( 1 + (-0.277 - 4.11i)T \) |
| good | 3 | \( 1 + (-3.07 + 0.739i)T + (2.67 - 1.36i)T^{2} \) |
| 5 | \( 1 + (0.951 + 0.0748i)T + (4.93 + 0.782i)T^{2} \) |
| 7 | \( 1 + (0.0553 - 0.230i)T + (-6.23 - 3.17i)T^{2} \) |
| 13 | \( 1 + (1.35 + 0.439i)T + (10.5 + 7.64i)T^{2} \) |
| 19 | \( 1 + (0.637 - 4.02i)T + (-18.0 - 5.87i)T^{2} \) |
| 23 | \( 1 + (4.69 - 1.94i)T + (16.2 - 16.2i)T^{2} \) |
| 29 | \( 1 + (2.08 - 1.27i)T + (13.1 - 25.8i)T^{2} \) |
| 31 | \( 1 + (-1.51 + 1.77i)T + (-4.84 - 30.6i)T^{2} \) |
| 37 | \( 1 + (3.02 + 4.92i)T + (-16.7 + 32.9i)T^{2} \) |
| 41 | \( 1 + (5.89 + 3.61i)T + (18.6 + 36.5i)T^{2} \) |
| 43 | \( 1 + (-1.15 - 1.15i)T + 43iT^{2} \) |
| 47 | \( 1 + (4.26 + 5.87i)T + (-14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (4.58 + 9.00i)T + (-31.1 + 42.8i)T^{2} \) |
| 59 | \( 1 + (-2.29 - 14.5i)T + (-56.1 + 18.2i)T^{2} \) |
| 61 | \( 1 + (-2.45 + 2.09i)T + (9.54 - 60.2i)T^{2} \) |
| 67 | \( 1 - 6.41T + 67T^{2} \) |
| 71 | \( 1 + (-0.801 + 10.1i)T + (-70.1 - 11.1i)T^{2} \) |
| 73 | \( 1 + (-0.437 + 0.268i)T + (33.1 - 65.0i)T^{2} \) |
| 79 | \( 1 + (-1.00 - 12.8i)T + (-78.0 + 12.3i)T^{2} \) |
| 83 | \( 1 + (5.16 + 2.63i)T + (48.7 + 67.1i)T^{2} \) |
| 89 | \( 1 + 5.06iT - 89T^{2} \) |
| 97 | \( 1 + (-7.72 - 6.59i)T + (15.1 + 95.8i)T^{2} \) |
| 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
−11.83888482001416303823037945294, −10.20714001625791343217089300698, −9.250516415440341823208943480208, −8.348535490657997127015278089256, −7.938700723127719310373162748793, −6.94995002091272428705060945518, −5.83741032806315367936617732059, −3.97887342032068533143857562994, −3.53892217924628497386271630332, −1.95302708742333440790817806277,
1.97210801106841976063886770373, 3.03914965714347359288255402001, 4.05678204970361866006600686551, 4.79986011713870362863016521953, 6.82761906334989120628357497585, 7.77522784881813791296533517661, 8.730366580968541877282652815310, 9.596701884029727457952611475561, 10.04031344287158982382642410249, 11.37718734000444239502153075627
