| L(s) = 1 | + (1.88 − 1.37i)2-s + (−0.648 + 1.56i)3-s + (1.06 − 3.27i)4-s + (0.944 − 2.02i)5-s + (0.923 + 3.84i)6-s + (−0.489 − 0.799i)7-s + (−1.03 − 3.19i)8-s + (0.0884 + 0.0884i)9-s + (−0.996 − 5.11i)10-s + (−0.652 − 0.763i)11-s + (4.43 + 3.78i)12-s + (0.806 + 3.35i)13-s + (−2.02 − 0.836i)14-s + (2.56 + 2.79i)15-s + (−0.774 − 0.563i)16-s + (−3.50 + 2.98i)17-s + ⋯ |
| L(s) = 1 | + (1.33 − 0.969i)2-s + (−0.374 + 0.904i)3-s + (0.531 − 1.63i)4-s + (0.422 − 0.906i)5-s + (0.376 + 1.57i)6-s + (−0.185 − 0.302i)7-s + (−0.367 − 1.13i)8-s + (0.0294 + 0.0294i)9-s + (−0.314 − 1.61i)10-s + (−0.196 − 0.230i)11-s + (1.28 + 1.09i)12-s + (0.223 + 0.931i)13-s + (−0.539 − 0.223i)14-s + (0.661 + 0.721i)15-s + (−0.193 − 0.140i)16-s + (−0.849 + 0.725i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.556 + 0.830i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.556 + 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.89006 - 1.00847i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.89006 - 1.00847i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-0.944 + 2.02i)T \) |
| 41 | \( 1 + (-6.23 + 1.45i)T \) |
| good | 2 | \( 1 + (-1.88 + 1.37i)T + (0.618 - 1.90i)T^{2} \) |
| 3 | \( 1 + (0.648 - 1.56i)T + (-2.12 - 2.12i)T^{2} \) |
| 7 | \( 1 + (0.489 + 0.799i)T + (-3.17 + 6.23i)T^{2} \) |
| 11 | \( 1 + (0.652 + 0.763i)T + (-1.72 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-0.806 - 3.35i)T + (-11.5 + 5.90i)T^{2} \) |
| 17 | \( 1 + (3.50 - 2.98i)T + (2.65 - 16.7i)T^{2} \) |
| 19 | \( 1 + (2.39 + 3.91i)T + (-8.62 + 16.9i)T^{2} \) |
| 23 | \( 1 + (-0.0566 + 0.357i)T + (-21.8 - 7.10i)T^{2} \) |
| 29 | \( 1 + (-0.530 - 6.73i)T + (-28.6 + 4.53i)T^{2} \) |
| 31 | \( 1 + (6.65 - 2.16i)T + (25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-3.22 - 6.32i)T + (-21.7 + 29.9i)T^{2} \) |
| 43 | \( 1 + (3.68 + 5.07i)T + (-13.2 + 40.8i)T^{2} \) |
| 47 | \( 1 + (0.376 + 0.230i)T + (21.3 + 41.8i)T^{2} \) |
| 53 | \( 1 + (-1.48 + 1.73i)T + (-8.29 - 52.3i)T^{2} \) |
| 59 | \( 1 + (1.13 + 1.55i)T + (-18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (2.21 + 13.9i)T + (-58.0 + 18.8i)T^{2} \) |
| 67 | \( 1 + (-7.05 + 0.554i)T + (66.1 - 10.4i)T^{2} \) |
| 71 | \( 1 + (-6.39 + 5.46i)T + (11.1 - 70.1i)T^{2} \) |
| 73 | \( 1 - 10.3iT - 73T^{2} \) |
| 79 | \( 1 + (3.00 + 7.24i)T + (-55.8 + 55.8i)T^{2} \) |
| 83 | \( 1 + (-1.77 - 1.77i)T + 83iT^{2} \) |
| 89 | \( 1 + (0.243 + 0.0584i)T + (79.2 + 40.4i)T^{2} \) |
| 97 | \( 1 + (0.0873 + 1.10i)T + (-95.8 + 15.1i)T^{2} \) |
| 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
−12.43252405631464101605371247385, −11.19520852874312957653709608939, −10.74216921481911915666099206205, −9.681156472416491897248706680621, −8.662324949751451318758513871393, −6.58312452879055518208082036737, −5.33107401208311066693858875149, −4.60734063471983483904891383068, −3.77486640579467341467784195908, −1.92785063445960919131935333754,
2.54816252617582937346862819811, 4.03665315216662947686325732537, 5.68878528667575094813062937363, 6.17150805983490423709305322589, 7.14301395282160305127871812169, 7.83915324492457373511525200192, 9.637342964986370325423515851201, 10.96787005219634001383182111598, 12.05344808223149840679426920352, 12.94413077933047512765054737120
