| 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 | \( 1 + (-5.29 + 5.29i)T - 97iT^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.29388746631701704615143915270, −10.03561569772244890913637371036, −9.038286540843047127372283854896, −8.198662644073964845658290519756, −7.39714764404583500485108978074, −5.96176043745198973468753240082, −5.56344359665229952501224066862, −4.26475050710028225344977535317, −2.82497018463251385371873440786, −1.29007751983184376501693729930,
1.17570396891785438474826199840, 2.89528778815132111195055132553, 4.52642668358648728641540958700, 4.97703320110833818520219370877, 6.19097923889736225037192515012, 7.46309162368222724170986678402, 8.045395099356732904663668616253, 9.073651479018263974365818250236, 10.36420756566207390480514467366, 10.81241730260420937280246718784
