| L(s) = 1 | + (−0.210 − 0.412i)2-s + (1.44 − 0.958i)3-s + (1.04 − 1.44i)4-s + (0.654 + 4.13i)5-s + (−0.698 − 0.393i)6-s + (−0.663 + 0.566i)7-s + (−1.73 − 0.274i)8-s + (1.16 − 2.76i)9-s + (1.56 − 1.13i)10-s + (−2.71 − 1.66i)11-s + (0.130 − 3.09i)12-s + (−4.33 + 0.341i)13-s + (0.372 + 0.154i)14-s + (4.90 + 5.33i)15-s + (−0.853 − 2.62i)16-s + (−0.884 + 0.212i)17-s + ⋯ |
| L(s) = 1 | + (−0.148 − 0.291i)2-s + (0.833 − 0.553i)3-s + (0.524 − 0.722i)4-s + (0.292 + 1.84i)5-s + (−0.285 − 0.160i)6-s + (−0.250 + 0.214i)7-s + (−0.611 − 0.0969i)8-s + (0.388 − 0.921i)9-s + (0.495 − 0.359i)10-s + (−0.820 − 0.502i)11-s + (0.0376 − 0.892i)12-s + (−1.20 + 0.0946i)13-s + (0.0996 + 0.0412i)14-s + (1.26 + 1.37i)15-s + (−0.213 − 0.656i)16-s + (−0.214 + 0.0515i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 123 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.849 + 0.528i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 123 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.849 + 0.528i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.26243 - 0.360586i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.26243 - 0.360586i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.44 + 0.958i)T \) |
| 41 | \( 1 + (-1.19 + 6.29i)T \) |
| good | 2 | \( 1 + (0.210 + 0.412i)T + (-1.17 + 1.61i)T^{2} \) |
| 5 | \( 1 + (-0.654 - 4.13i)T + (-4.75 + 1.54i)T^{2} \) |
| 7 | \( 1 + (0.663 - 0.566i)T + (1.09 - 6.91i)T^{2} \) |
| 11 | \( 1 + (2.71 + 1.66i)T + (4.99 + 9.80i)T^{2} \) |
| 13 | \( 1 + (4.33 - 0.341i)T + (12.8 - 2.03i)T^{2} \) |
| 17 | \( 1 + (0.884 - 0.212i)T + (15.1 - 7.71i)T^{2} \) |
| 19 | \( 1 + (-4.24 - 0.334i)T + (18.7 + 2.97i)T^{2} \) |
| 23 | \( 1 + (0.0470 - 0.144i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (2.47 + 0.593i)T + (25.8 + 13.1i)T^{2} \) |
| 31 | \( 1 + (-2.54 - 3.50i)T + (-9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-7.30 - 5.30i)T + (11.4 + 35.1i)T^{2} \) |
| 43 | \( 1 + (-4.24 + 2.16i)T + (25.2 - 34.7i)T^{2} \) |
| 47 | \( 1 + (-2.90 + 3.39i)T + (-7.35 - 46.4i)T^{2} \) |
| 53 | \( 1 + (2.15 - 8.96i)T + (-47.2 - 24.0i)T^{2} \) |
| 59 | \( 1 + (-5.54 - 1.80i)T + (47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (-5.18 + 10.1i)T + (-35.8 - 49.3i)T^{2} \) |
| 67 | \( 1 + (1.26 - 0.775i)T + (30.4 - 59.6i)T^{2} \) |
| 71 | \( 1 + (-3.04 + 4.96i)T + (-32.2 - 63.2i)T^{2} \) |
| 73 | \( 1 + (5.10 + 5.10i)T + 73iT^{2} \) |
| 79 | \( 1 + (3.13 - 1.29i)T + (55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 - 6.73iT - 83T^{2} \) |
| 89 | \( 1 + (4.10 + 4.80i)T + (-13.9 + 87.9i)T^{2} \) |
| 97 | \( 1 + (-1.82 - 2.97i)T + (-44.0 + 86.4i)T^{2} \) |
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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
−13.67002638230803529043657557699, −12.19786877453083668111769289117, −11.12791925253736028268026947115, −10.17493733372503218021205569786, −9.456338134407490828310958998359, −7.64058410088306865404606453184, −6.87594306855878648703672009354, −5.82777020563007120942534182671, −3.05787074433560204159524974956, −2.37890470644099460121113799959,
2.46226332221243273219859838065, 4.27452366427808116285918108050, 5.36914963372050459797324852327, 7.45493021591467685250194043415, 8.144222999300983639955298212479, 9.253284770334073351238668041356, 9.916371293635257229419632999670, 11.66688037839373729069839091182, 12.85911490787057971579744367288, 13.15506501451826156480816098036
