| L(s) = 1 | + (−0.857 + 1.17i)2-s + (−0.707 − 2.17i)3-s + (−0.0390 − 0.120i)4-s + (−1.58 − 1.58i)5-s + (3.17 + 1.03i)6-s + 2.12·7-s + (−2.59 − 0.844i)8-s + (−1.80 + 1.31i)9-s + (3.22 − 0.510i)10-s + (−0.204 + 0.281i)11-s + (−0.233 + 0.169i)12-s + (−0.627 − 0.863i)13-s + (−1.81 + 2.50i)14-s + (−2.32 + 4.55i)15-s + (3.42 − 2.49i)16-s + (−3.99 − 1.00i)17-s + ⋯ |
| L(s) = 1 | + (−0.606 + 0.834i)2-s + (−0.408 − 1.25i)3-s + (−0.0195 − 0.0600i)4-s + (−0.707 − 0.706i)5-s + (1.29 + 0.420i)6-s + 0.801·7-s + (−0.918 − 0.298i)8-s + (−0.602 + 0.437i)9-s + (1.01 − 0.161i)10-s + (−0.0615 + 0.0847i)11-s + (−0.0674 + 0.0490i)12-s + (−0.173 − 0.239i)13-s + (−0.485 + 0.668i)14-s + (−0.599 + 1.17i)15-s + (0.856 − 0.622i)16-s + (−0.969 − 0.244i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.841 + 0.539i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.841 + 0.539i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0775708 - 0.264844i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0775708 - 0.264844i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (1.58 + 1.58i)T \) |
| 17 | \( 1 + (3.99 + 1.00i)T \) |
| good | 2 | \( 1 + (0.857 - 1.17i)T + (-0.618 - 1.90i)T^{2} \) |
| 3 | \( 1 + (0.707 + 2.17i)T + (-2.42 + 1.76i)T^{2} \) |
| 7 | \( 1 - 2.12T + 7T^{2} \) |
| 11 | \( 1 + (0.204 - 0.281i)T + (-3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (0.627 + 0.863i)T + (-4.01 + 12.3i)T^{2} \) |
| 19 | \( 1 + (1.34 - 4.12i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (3.42 + 2.48i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (4.63 - 1.50i)T + (23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (9.91 + 3.22i)T + (25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-4.75 + 3.45i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (-2.15 - 2.96i)T + (-12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 3.26iT - 43T^{2} \) |
| 47 | \( 1 + (10.2 - 3.33i)T + (38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-2.85 + 0.927i)T + (42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (-9.15 + 6.64i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (0.507 - 0.698i)T + (-18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (-6.30 - 2.04i)T + (54.2 + 39.3i)T^{2} \) |
| 71 | \( 1 + (-13.0 + 4.23i)T + (57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (9.94 + 7.22i)T + (22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (8.36 - 2.71i)T + (63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-6.43 - 2.09i)T + (67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (3.90 + 2.83i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (0.345 + 1.06i)T + (-78.4 + 57.0i)T^{2} \) |
| 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
−11.16849473704831492786282640885, −9.534249915019401403117577379900, −8.516834210937453761617282115402, −7.86627230752493785557778856650, −7.36163536356489834828724676862, −6.36956386061580386472289667328, −5.37194877763034155742604829399, −3.95092601164463662401379414761, −1.90516600978241298878923770389, −0.21507155124656342556480430185,
2.13166646763617227117083723395, 3.56762470185330007631095423170, 4.53945358137746384578016817266, 5.63667926478292726587074029906, 6.95284543119197326892886182704, 8.222835117486237361593868801749, 9.169920948272353081474653790110, 9.960336987378094800052039190719, 10.83342138361281322908333423542, 11.24083593846548332984145766007
