| L(s) = 1 | + (0.642 − 0.766i)2-s + (1.59 + 0.670i)3-s + (−0.173 − 0.984i)4-s + (0.906 − 2.04i)5-s + (1.54 − 0.792i)6-s + (−0.609 + 0.351i)7-s + (−0.866 − 0.500i)8-s + (2.10 + 2.14i)9-s + (−0.983 − 2.00i)10-s + (−0.509 − 0.294i)11-s + (0.383 − 1.68i)12-s + (4.66 + 1.69i)13-s + (−0.122 + 0.692i)14-s + (2.81 − 2.65i)15-s + (−0.939 + 0.342i)16-s + (−2.98 − 2.50i)17-s + ⋯ |
| L(s) = 1 | + (0.454 − 0.541i)2-s + (0.921 + 0.387i)3-s + (−0.0868 − 0.492i)4-s + (0.405 − 0.914i)5-s + (0.628 − 0.323i)6-s + (−0.230 + 0.132i)7-s + (−0.306 − 0.176i)8-s + (0.700 + 0.714i)9-s + (−0.310 − 0.635i)10-s + (−0.153 − 0.0887i)11-s + (0.110 − 0.487i)12-s + (1.29 + 0.470i)13-s + (−0.0326 + 0.185i)14-s + (0.727 − 0.685i)15-s + (−0.234 + 0.0855i)16-s + (−0.724 − 0.607i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.582 + 0.812i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.582 + 0.812i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.31992 - 1.19082i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.31992 - 1.19082i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.642 + 0.766i)T \) |
| 3 | \( 1 + (-1.59 - 0.670i)T \) |
| 5 | \( 1 + (-0.906 + 2.04i)T \) |
| 19 | \( 1 + (-4.25 + 0.930i)T \) |
| good | 7 | \( 1 + (0.609 - 0.351i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (0.509 + 0.294i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-4.66 - 1.69i)T + (9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (2.98 + 2.50i)T + (2.95 + 16.7i)T^{2} \) |
| 23 | \( 1 + (0.920 + 5.21i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (5.26 - 4.42i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (3.31 - 1.91i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 4.06T + 37T^{2} \) |
| 41 | \( 1 + (11.2 - 4.09i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (-6.70 - 1.18i)T + (40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (-3.96 + 3.32i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + (1.73 - 0.305i)T + (49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (-6.40 - 5.37i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-0.127 - 0.722i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (9.62 - 8.07i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (0.366 - 2.07i)T + (-66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (2.97 + 8.17i)T + (-55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (-4.37 - 12.0i)T + (-60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-6.11 - 10.5i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (8.31 + 3.02i)T + (68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (8.74 + 7.33i)T + (16.8 + 95.5i)T^{2} \) |
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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
−10.59046071673923117928588993413, −9.567149250899377151425614979235, −9.018157457118326388467294217114, −8.342242415788395719920671421288, −6.98135857731979749049743078679, −5.73230953732902314276210226855, −4.75472063497187391287899784308, −3.86903816166623951243486840379, −2.71883235008134987206307079274, −1.45509475999341183885519356794,
1.92018860785652839728218715224, 3.27968348289454122471349407811, 3.86603330968973771063410580087, 5.64198490428321354805994028736, 6.37469521606392409547401123065, 7.32457424697698457401802491938, 7.958501890771577033048537779393, 9.023497645922575833902607496454, 9.834782842567954723654809724951, 10.86125238493712171922292727025
