| L(s) = 1 | + (−0.743 + 0.669i)2-s + (−0.429 + 1.67i)3-s + (0.104 − 0.994i)4-s + (0.680 − 2.12i)5-s + (−0.803 − 1.53i)6-s + (−0.636 − 0.367i)7-s + (0.587 + 0.809i)8-s + (−2.63 − 1.44i)9-s + (0.919 + 2.03i)10-s + (1.82 + 2.02i)11-s + (1.62 + 0.602i)12-s + (1.59 + 1.43i)13-s + (0.718 − 0.152i)14-s + (3.28 + 2.05i)15-s + (−0.978 − 0.207i)16-s + (4.42 + 6.08i)17-s + ⋯ |
| L(s) = 1 | + (−0.525 + 0.473i)2-s + (−0.248 + 0.968i)3-s + (0.0522 − 0.497i)4-s + (0.304 − 0.952i)5-s + (−0.328 − 0.626i)6-s + (−0.240 − 0.138i)7-s + (0.207 + 0.286i)8-s + (−0.876 − 0.480i)9-s + (0.290 + 0.644i)10-s + (0.549 + 0.609i)11-s + (0.468 + 0.173i)12-s + (0.443 + 0.399i)13-s + (0.192 − 0.0408i)14-s + (0.847 + 0.531i)15-s + (−0.244 − 0.0519i)16-s + (1.07 + 1.47i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.305 - 0.952i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.305 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.849360 + 0.619642i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.849360 + 0.619642i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.743 - 0.669i)T \) |
| 3 | \( 1 + (0.429 - 1.67i)T \) |
| 5 | \( 1 + (-0.680 + 2.12i)T \) |
| good | 7 | \( 1 + (0.636 + 0.367i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.82 - 2.02i)T + (-1.14 + 10.9i)T^{2} \) |
| 13 | \( 1 + (-1.59 - 1.43i)T + (1.35 + 12.9i)T^{2} \) |
| 17 | \( 1 + (-4.42 - 6.08i)T + (-5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (-5.79 + 4.20i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (-0.786 - 3.70i)T + (-21.0 + 9.35i)T^{2} \) |
| 29 | \( 1 + (6.14 - 2.73i)T + (19.4 - 21.5i)T^{2} \) |
| 31 | \( 1 + (-7.30 - 3.25i)T + (20.7 + 23.0i)T^{2} \) |
| 37 | \( 1 + (-3.70 + 1.20i)T + (29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (4.07 - 4.52i)T + (-4.28 - 40.7i)T^{2} \) |
| 43 | \( 1 + (-5.20 - 3.00i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (0.0735 + 0.165i)T + (-31.4 + 34.9i)T^{2} \) |
| 53 | \( 1 + (2.88 - 3.97i)T + (-16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-6.23 + 6.92i)T + (-6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 + (8.14 + 9.04i)T + (-6.37 + 60.6i)T^{2} \) |
| 67 | \( 1 + (2.02 - 4.55i)T + (-44.8 - 49.7i)T^{2} \) |
| 71 | \( 1 + (-5.00 - 3.63i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-2.79 - 0.909i)T + (59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (0.894 - 0.398i)T + (52.8 - 58.7i)T^{2} \) |
| 83 | \( 1 + (-5.44 + 0.572i)T + (81.1 - 17.2i)T^{2} \) |
| 89 | \( 1 + (-5.39 + 16.5i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (5.80 + 13.0i)T + (-64.9 + 72.0i)T^{2} \) |
| 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
−11.13840173799724588523741703238, −9.966511042615241792575883050341, −9.541086904742985508202106632849, −8.786416759087620242250663648578, −7.80241129023493515003687628811, −6.47121412945199859450208110300, −5.56600865873163308595495971748, −4.69494398514829343323005294236, −3.53570613424708465381390775613, −1.31257634513072592185158720409,
0.998502265868531393878240751785, 2.59343552028674533176592695354, 3.43658244405062657606892005489, 5.54674567604375986943817774431, 6.32397296719194412826754269617, 7.38118319309629392094029145581, 7.980565166890393397398627135526, 9.253119166951889907952208764504, 10.04646459926214600032825039385, 11.05173570101994413112697399843
