| L(s) = 1 | + (−0.173 + 0.984i)2-s + (−1.17 − 0.984i)3-s + (−0.939 − 0.342i)4-s + (−1.87 + 0.684i)5-s + (1.17 − 0.984i)6-s + (−1.34 − 2.33i)7-s + (0.5 − 0.866i)8-s + (−0.113 − 0.642i)9-s + (−0.347 − 1.96i)10-s + (−1.59 + 2.75i)11-s + (0.766 + 1.32i)12-s + (4.41 − 3.70i)13-s + (2.53 − 0.921i)14-s + (2.87 + 1.04i)15-s + (0.766 + 0.642i)16-s + (−1.13 + 6.41i)17-s + ⋯ |
| L(s) = 1 | + (−0.122 + 0.696i)2-s + (−0.677 − 0.568i)3-s + (−0.469 − 0.171i)4-s + (−0.840 + 0.305i)5-s + (0.479 − 0.402i)6-s + (−0.509 − 0.882i)7-s + (0.176 − 0.306i)8-s + (−0.0377 − 0.214i)9-s + (−0.109 − 0.622i)10-s + (−0.480 + 0.831i)11-s + (0.221 + 0.383i)12-s + (1.22 − 1.02i)13-s + (0.676 − 0.246i)14-s + (0.743 + 0.270i)15-s + (0.191 + 0.160i)16-s + (−0.274 + 1.55i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.216 - 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.216 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.317562 + 0.395728i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.317562 + 0.395728i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.173 - 0.984i)T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + (1.17 + 0.984i)T + (0.520 + 2.95i)T^{2} \) |
| 5 | \( 1 + (1.87 - 0.684i)T + (3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (1.34 + 2.33i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (1.59 - 2.75i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-4.41 + 3.70i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (1.13 - 6.41i)T + (-15.9 - 5.81i)T^{2} \) |
| 23 | \( 1 + (-0.652 - 0.237i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (-0.490 - 2.78i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (-1.22 - 2.12i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 4.36T + 37T^{2} \) |
| 41 | \( 1 + (-0.266 - 0.223i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (5.69 - 2.07i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-1.36 - 7.76i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (-7.71 - 2.80i)T + (40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (0.0996 - 0.565i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (2.75 + 1.00i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-0.860 - 4.88i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (7.94 - 2.89i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (-12.0 - 10.1i)T + (12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (-6.94 - 5.82i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (4.23 + 7.34i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (5.92 - 4.97i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (-0.0603 + 0.342i)T + (-91.1 - 33.1i)T^{2} \) |
| 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
−10.67147253612424174901790983397, −9.939265842373298154712759782257, −8.620099848406075766963354321682, −7.83396464447511416649371123570, −7.10611243607851481933556887877, −6.38971977861673321472821714507, −5.60257475697543798172921021418, −4.20067179688442795838569253861, −3.40957141468682619070331571948, −1.10696504641747476647113529027,
0.36437164730465586325549779177, 2.42973541215523295422982346724, 3.62759081793717797085202108991, 4.57606739346742302990513698457, 5.47583786335679146200404841454, 6.41762506945115936436636837932, 7.83170990579324728303874592896, 8.686269204285642783888558810848, 9.324502825772628005203463916692, 10.31405608234595730256222776158
