| L(s) = 1 | + (−0.142 + 0.989i)2-s + (−0.0488 − 0.106i)3-s + (−0.959 − 0.281i)4-s + (0.654 + 0.755i)5-s + (0.112 − 0.0331i)6-s + (1.43 + 0.919i)7-s + (0.415 − 0.909i)8-s + (1.95 − 2.25i)9-s + (−0.841 + 0.540i)10-s + (0.922 + 6.41i)11-s + (0.0167 + 0.116i)12-s + (−0.840 + 0.540i)13-s + (−1.11 + 1.28i)14-s + (0.0488 − 0.106i)15-s + (0.841 + 0.540i)16-s + (−4.80 + 1.40i)17-s + ⋯ |
| L(s) = 1 | + (−0.100 + 0.699i)2-s + (−0.0282 − 0.0617i)3-s + (−0.479 − 0.140i)4-s + (0.292 + 0.337i)5-s + (0.0460 − 0.0135i)6-s + (0.540 + 0.347i)7-s + (0.146 − 0.321i)8-s + (0.651 − 0.752i)9-s + (−0.266 + 0.170i)10-s + (0.278 + 1.93i)11-s + (0.00483 + 0.0336i)12-s + (−0.233 + 0.149i)13-s + (−0.297 + 0.343i)14-s + (0.0126 − 0.0276i)15-s + (0.210 + 0.135i)16-s + (−1.16 + 0.341i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.235 - 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.235 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.969287 + 0.762152i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.969287 + 0.762152i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.142 - 0.989i)T \) |
| 5 | \( 1 + (-0.654 - 0.755i)T \) |
| 23 | \( 1 + (-3.59 + 3.17i)T \) |
| good | 3 | \( 1 + (0.0488 + 0.106i)T + (-1.96 + 2.26i)T^{2} \) |
| 7 | \( 1 + (-1.43 - 0.919i)T + (2.90 + 6.36i)T^{2} \) |
| 11 | \( 1 + (-0.922 - 6.41i)T + (-10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (0.840 - 0.540i)T + (5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (4.80 - 1.40i)T + (14.3 - 9.19i)T^{2} \) |
| 19 | \( 1 + (-3.14 - 0.924i)T + (15.9 + 10.2i)T^{2} \) |
| 29 | \( 1 + (0.346 - 0.101i)T + (24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (-3.99 + 8.74i)T + (-20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 + (5.41 - 6.24i)T + (-5.26 - 36.6i)T^{2} \) |
| 41 | \( 1 + (2.84 + 3.28i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (3.89 + 8.53i)T + (-28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 - 0.735T + 47T^{2} \) |
| 53 | \( 1 + (-0.0471 - 0.0302i)T + (22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (-12.0 + 7.75i)T + (24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (-4.22 + 9.24i)T + (-39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (-0.568 + 3.95i)T + (-64.2 - 18.8i)T^{2} \) |
| 71 | \( 1 + (1.52 - 10.6i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (8.05 + 2.36i)T + (61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (6.62 - 4.26i)T + (32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (6.26 - 7.23i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (-3.35 - 7.35i)T + (-58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 + (9.33 + 10.7i)T + (-13.8 + 96.0i)T^{2} \) |
| 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
−12.47401034937736950297561460084, −11.59137750535350315500161001088, −10.12413997719188089755600716079, −9.548345553459086487625932107895, −8.434847112411375308356478150906, −7.08255994939075352047801006658, −6.68193475752974967342326493510, −5.13359331487247501535155884933, −4.14230075516137603960150927031, −1.97845377242409285216908032505,
1.28470590325348082458272292404, 3.05026579558718208288657765468, 4.52275519947408282459976764453, 5.49665666451488683845224492622, 7.08290532393018188987905439015, 8.336418345230749434652351287477, 9.067083958334025927937188884184, 10.28835413471006921683766512720, 11.07160317820732140519145028369, 11.72482033629412424469492454584
