| L(s) = 1 | + (1.34 − 0.436i)2-s + (0.296 + 1.97i)3-s + (1.61 − 1.17i)4-s + (0.830 + 0.326i)5-s + (1.25 + 2.52i)6-s + (1.00 − 2.44i)7-s + (1.66 − 2.28i)8-s + (−0.927 + 0.286i)9-s + (1.26 + 0.0760i)10-s + (−0.0695 + 0.225i)11-s + (2.79 + 2.84i)12-s + (−1.87 + 0.428i)13-s + (0.287 − 3.73i)14-s + (−0.395 + 1.73i)15-s + (1.24 − 3.80i)16-s + (−5.72 + 3.90i)17-s + ⋯ |
| L(s) = 1 | + (0.951 − 0.308i)2-s + (0.171 + 1.13i)3-s + (0.809 − 0.587i)4-s + (0.371 + 0.145i)5-s + (0.514 + 1.02i)6-s + (0.380 − 0.924i)7-s + (0.588 − 0.808i)8-s + (−0.309 + 0.0953i)9-s + (0.398 + 0.0240i)10-s + (−0.0209 + 0.0680i)11-s + (0.806 + 0.820i)12-s + (−0.520 + 0.118i)13-s + (0.0769 − 0.997i)14-s + (−0.102 + 0.447i)15-s + (0.310 − 0.950i)16-s + (−1.38 + 0.947i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 - 0.201i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.979 - 0.201i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.65834 + 0.270007i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.65834 + 0.270007i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.34 + 0.436i)T \) |
| 7 | \( 1 + (-1.00 + 2.44i)T \) |
| good | 3 | \( 1 + (-0.296 - 1.97i)T + (-2.86 + 0.884i)T^{2} \) |
| 5 | \( 1 + (-0.830 - 0.326i)T + (3.66 + 3.40i)T^{2} \) |
| 11 | \( 1 + (0.0695 - 0.225i)T + (-9.08 - 6.19i)T^{2} \) |
| 13 | \( 1 + (1.87 - 0.428i)T + (11.7 - 5.64i)T^{2} \) |
| 17 | \( 1 + (5.72 - 3.90i)T + (6.21 - 15.8i)T^{2} \) |
| 19 | \( 1 + (2.06 - 1.19i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-5.14 - 3.50i)T + (8.40 + 21.4i)T^{2} \) |
| 29 | \( 1 + (-0.663 + 1.37i)T + (-18.0 - 22.6i)T^{2} \) |
| 31 | \( 1 + (-0.282 + 0.488i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (11.9 - 0.895i)T + (36.5 - 5.51i)T^{2} \) |
| 41 | \( 1 + (3.07 + 3.85i)T + (-9.12 + 39.9i)T^{2} \) |
| 43 | \( 1 + (-4.72 - 3.76i)T + (9.56 + 41.9i)T^{2} \) |
| 47 | \( 1 + (-4.25 + 3.94i)T + (3.51 - 46.8i)T^{2} \) |
| 53 | \( 1 + (1.27 + 0.0956i)T + (52.4 + 7.89i)T^{2} \) |
| 59 | \( 1 + (7.44 - 2.92i)T + (43.2 - 40.1i)T^{2} \) |
| 61 | \( 1 + (-12.7 + 0.953i)T + (60.3 - 9.09i)T^{2} \) |
| 67 | \( 1 + (5.72 + 3.30i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-6.57 + 3.16i)T + (44.2 - 55.5i)T^{2} \) |
| 73 | \( 1 + (7.04 + 6.53i)T + (5.45 + 72.7i)T^{2} \) |
| 79 | \( 1 + (0.456 + 0.790i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-6.00 - 1.37i)T + (74.7 + 36.0i)T^{2} \) |
| 89 | \( 1 + (-8.44 + 2.60i)T + (73.5 - 50.1i)T^{2} \) |
| 97 | \( 1 - 4.64T + 97T^{2} \) |
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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
−11.07286154968134830560291711203, −10.53930499466280655610647698637, −9.925563329142306679465835843862, −8.825282497073678046066939258150, −7.34627517560394040110247387014, −6.41759385428379667812079111038, −5.10064516388039769221862530467, −4.32746880007207104405044489326, −3.54597346360562594192245369406, −1.95970610270098724076492571673,
1.94242693857319925437403281049, 2.75405694136908559253037005398, 4.62025171951717996030798819651, 5.47437637076442330450820251170, 6.64180315664708067728332037231, 7.18556433013542050948212945702, 8.325084196845179322707239104171, 9.111283100978926484009752277314, 10.73219052660197557039357885132, 11.66363852337486892677191305238
