| L(s) = 1 | + (1.23 + 0.577i)2-s + (−0.618 − 1.61i)3-s + (−0.0838 − 0.0999i)4-s + (2.03 + 0.927i)5-s + (0.169 − 2.36i)6-s + (2.37 + 0.636i)7-s + (−0.753 − 2.81i)8-s + (−2.23 + 2.00i)9-s + (1.98 + 2.32i)10-s + (−1.15 − 0.664i)11-s + (−0.109 + 0.197i)12-s + (2.87 − 4.10i)13-s + (2.57 + 2.16i)14-s + (0.243 − 3.86i)15-s + (0.646 − 3.66i)16-s + (−3.01 − 1.40i)17-s + ⋯ |
| L(s) = 1 | + (0.876 + 0.408i)2-s + (−0.356 − 0.934i)3-s + (−0.0419 − 0.0499i)4-s + (0.909 + 0.414i)5-s + (0.0690 − 0.964i)6-s + (0.897 + 0.240i)7-s + (−0.266 − 0.994i)8-s + (−0.745 + 0.666i)9-s + (0.627 + 0.735i)10-s + (−0.347 − 0.200i)11-s + (−0.0317 + 0.0569i)12-s + (0.797 − 1.13i)13-s + (0.688 + 0.577i)14-s + (0.0628 − 0.998i)15-s + (0.161 − 0.916i)16-s + (−0.732 − 0.341i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.862 + 0.505i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.862 + 0.505i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.88328 - 0.510877i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.88328 - 0.510877i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.618 + 1.61i)T \) |
| 5 | \( 1 + (-2.03 - 0.927i)T \) |
| 19 | \( 1 + (-2.95 - 3.20i)T \) |
| good | 2 | \( 1 + (-1.23 - 0.577i)T + (1.28 + 1.53i)T^{2} \) |
| 7 | \( 1 + (-2.37 - 0.636i)T + (6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (1.15 + 0.664i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.87 + 4.10i)T + (-4.44 - 12.2i)T^{2} \) |
| 17 | \( 1 + (3.01 + 1.40i)T + (10.9 + 13.0i)T^{2} \) |
| 23 | \( 1 + (0.447 - 5.12i)T + (-22.6 - 3.99i)T^{2} \) |
| 29 | \( 1 + (2.16 + 0.788i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (-0.564 - 0.977i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (7.02 - 7.02i)T - 37iT^{2} \) |
| 41 | \( 1 + (11.5 + 2.03i)T + (38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (0.177 + 2.03i)T + (-42.3 + 7.46i)T^{2} \) |
| 47 | \( 1 + (-4.79 - 10.2i)T + (-30.2 + 36.0i)T^{2} \) |
| 53 | \( 1 + (0.358 - 4.09i)T + (-52.1 - 9.20i)T^{2} \) |
| 59 | \( 1 + (-11.7 + 4.27i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (8.41 - 7.06i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (-6.98 + 3.25i)T + (43.0 - 51.3i)T^{2} \) |
| 71 | \( 1 + (-3.28 + 3.90i)T + (-12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (-0.203 + 0.142i)T + (24.9 - 68.5i)T^{2} \) |
| 79 | \( 1 + (6.77 + 1.19i)T + (74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (-5.91 - 1.58i)T + (71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + (0.552 + 3.13i)T + (-83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (-2.39 + 5.12i)T + (-62.3 - 74.3i)T^{2} \) |
| 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
−11.94830771009777853575433344902, −10.96570564241723787698998091681, −10.05031467253832886437345572886, −8.684920837081230598934236959703, −7.59519287483488965119715151497, −6.52146642065358433517517986403, −5.59986775983793086959339778394, −5.16456260217256601074715918914, −3.20248637903867564619375425092, −1.51601576699984304998860680770,
2.13853644336116680489142563883, 3.80926222509955999265541639265, 4.75126325210479788917683388041, 5.34639911547404650209788965629, 6.60628245932845348627198816600, 8.510859595785611061623989440914, 9.005565724160084772294475949389, 10.27442044531960243346022171460, 11.14879572158260409759210102279, 11.78724091835959420433714022157
