| L(s) = 1 | + (0.939 + 0.342i)2-s + (0.927 + 1.46i)3-s + (0.766 + 0.642i)4-s + (0.587 − 3.33i)5-s + (0.371 + 1.69i)6-s + (0.766 − 0.642i)7-s + (0.500 + 0.866i)8-s + (−1.27 + 2.71i)9-s + (1.69 − 2.93i)10-s + (0.344 + 1.95i)11-s + (−0.229 + 1.71i)12-s + (4.04 − 1.47i)13-s + (0.939 − 0.342i)14-s + (5.41 − 2.23i)15-s + (0.173 + 0.984i)16-s + (−2.73 + 4.72i)17-s + ⋯ |
| L(s) = 1 | + (0.664 + 0.241i)2-s + (0.535 + 0.844i)3-s + (0.383 + 0.321i)4-s + (0.262 − 1.49i)5-s + (0.151 + 0.690i)6-s + (0.289 − 0.242i)7-s + (0.176 + 0.306i)8-s + (−0.426 + 0.904i)9-s + (0.535 − 0.926i)10-s + (0.103 + 0.588i)11-s + (−0.0662 + 0.495i)12-s + (1.12 − 0.408i)13-s + (0.251 − 0.0914i)14-s + (1.39 − 0.576i)15-s + (0.0434 + 0.246i)16-s + (−0.662 + 1.14i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.861 - 0.507i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.861 - 0.507i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.34430 + 0.638897i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.34430 + 0.638897i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.939 - 0.342i)T \) |
| 3 | \( 1 + (-0.927 - 1.46i)T \) |
| 7 | \( 1 + (-0.766 + 0.642i)T \) |
| good | 5 | \( 1 + (-0.587 + 3.33i)T + (-4.69 - 1.71i)T^{2} \) |
| 11 | \( 1 + (-0.344 - 1.95i)T + (-10.3 + 3.76i)T^{2} \) |
| 13 | \( 1 + (-4.04 + 1.47i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (2.73 - 4.72i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3.16 + 5.47i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.24 - 1.04i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (8.55 + 3.11i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (-5.08 - 4.26i)T + (5.38 + 30.5i)T^{2} \) |
| 37 | \( 1 + (1.38 - 2.39i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (7.31 - 2.66i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (0.0470 + 0.266i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (-4.41 + 3.70i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + 12.3T + 53T^{2} \) |
| 59 | \( 1 + (-1.81 + 10.2i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (-1.50 + 1.26i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (-0.410 + 0.149i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (-5.21 + 9.03i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-2.40 - 4.15i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (3.80 + 1.38i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-12.2 - 4.45i)T + (63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + (-3.08 - 5.33i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (1.77 + 10.0i)T + (-91.1 + 33.1i)T^{2} \) |
| 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
−11.38255623057027882883928600457, −10.60679105136642681335178635384, −9.416070253569248346261417840164, −8.622464142024932096783578129145, −8.036141986807895170020809426302, −6.45516605773027202012165682917, −5.21563127045263187627961828490, −4.57205768034222525917488991328, −3.66351745672087832072678488194, −1.86959613155656120716895567628,
1.86931871058576259815598243587, 2.92966142532336346515363422814, 3.87577293784580753085702883130, 5.79482246922809120340438853984, 6.47230560711395932420553072418, 7.26893450366757119165105777591, 8.400831841543029680261457638134, 9.463769471198313433126154906010, 10.78007363826322975051011444924, 11.28209738921821711370227652961
