| L(s) = 1 | + (−0.731 + 0.391i)3-s + (−0.0771 − 0.0939i)5-s + (3.30 − 2.21i)7-s + (−1.28 + 1.92i)9-s + (−0.0222 + 0.00674i)11-s + (0.466 + 0.382i)13-s + (0.0931 + 0.0386i)15-s + (4.59 − 1.90i)17-s + (−0.166 − 1.68i)19-s + (−1.55 + 2.91i)21-s + (3.40 − 0.678i)23-s + (0.972 − 4.88i)25-s + (0.431 − 4.38i)27-s + (−1.44 + 4.77i)29-s + (6.39 + 6.39i)31-s + ⋯ |
| L(s) = 1 | + (−0.422 + 0.225i)3-s + (−0.0344 − 0.0420i)5-s + (1.25 − 0.835i)7-s + (−0.428 + 0.640i)9-s + (−0.00670 + 0.00203i)11-s + (0.129 + 0.106i)13-s + (0.0240 + 0.00996i)15-s + (1.11 − 0.461i)17-s + (−0.0381 − 0.386i)19-s + (−0.339 + 0.635i)21-s + (0.710 − 0.141i)23-s + (0.194 − 0.977i)25-s + (0.0831 − 0.844i)27-s + (−0.268 + 0.886i)29-s + (1.14 + 1.14i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0550i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0550i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.41164 - 0.0388587i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.41164 - 0.0388587i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + (0.731 - 0.391i)T + (1.66 - 2.49i)T^{2} \) |
| 5 | \( 1 + (0.0771 + 0.0939i)T + (-0.975 + 4.90i)T^{2} \) |
| 7 | \( 1 + (-3.30 + 2.21i)T + (2.67 - 6.46i)T^{2} \) |
| 11 | \( 1 + (0.0222 - 0.00674i)T + (9.14 - 6.11i)T^{2} \) |
| 13 | \( 1 + (-0.466 - 0.382i)T + (2.53 + 12.7i)T^{2} \) |
| 17 | \( 1 + (-4.59 + 1.90i)T + (12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (0.166 + 1.68i)T + (-18.6 + 3.70i)T^{2} \) |
| 23 | \( 1 + (-3.40 + 0.678i)T + (21.2 - 8.80i)T^{2} \) |
| 29 | \( 1 + (1.44 - 4.77i)T + (-24.1 - 16.1i)T^{2} \) |
| 31 | \( 1 + (-6.39 - 6.39i)T + 31iT^{2} \) |
| 37 | \( 1 + (-8.75 - 0.862i)T + (36.2 + 7.21i)T^{2} \) |
| 41 | \( 1 + (-0.0606 - 0.304i)T + (-37.8 + 15.6i)T^{2} \) |
| 43 | \( 1 + (-4.91 - 2.62i)T + (23.8 + 35.7i)T^{2} \) |
| 47 | \( 1 + (0.167 + 0.404i)T + (-33.2 + 33.2i)T^{2} \) |
| 53 | \( 1 + (-1.85 - 6.11i)T + (-44.0 + 29.4i)T^{2} \) |
| 59 | \( 1 + (7.95 - 6.53i)T + (11.5 - 57.8i)T^{2} \) |
| 61 | \( 1 + (6.59 + 12.3i)T + (-33.8 + 50.7i)T^{2} \) |
| 67 | \( 1 + (5.13 + 9.61i)T + (-37.2 + 55.7i)T^{2} \) |
| 71 | \( 1 + (3.99 + 5.97i)T + (-27.1 + 65.5i)T^{2} \) |
| 73 | \( 1 + (8.64 + 5.77i)T + (27.9 + 67.4i)T^{2} \) |
| 79 | \( 1 + (-0.504 + 1.21i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (16.2 - 1.60i)T + (81.4 - 16.1i)T^{2} \) |
| 89 | \( 1 + (-2.00 - 0.399i)T + (82.2 + 34.0i)T^{2} \) |
| 97 | \( 1 + (-5.29 - 5.29i)T + 97iT^{2} \) |
| 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
−10.81241730260420937280246718784, −10.36420756566207390480514467366, −9.073651479018263974365818250236, −8.045395099356732904663668616253, −7.46309162368222724170986678402, −6.19097923889736225037192515012, −4.97703320110833818520219370877, −4.52642668358648728641540958700, −2.89528778815132111195055132553, −1.17570396891785438474826199840,
1.29007751983184376501693729930, 2.82497018463251385371873440786, 4.26475050710028225344977535317, 5.56344359665229952501224066862, 5.96176043745198973468753240082, 7.39714764404583500485108978074, 8.198662644073964845658290519756, 9.038286540843047127372283854896, 10.03561569772244890913637371036, 11.29388746631701704615143915270
