| L(s) = 1 | + (−0.707 − 1.22i)2-s + (−0.999 + 1.73i)4-s + (5.12 − 2.95i)5-s + 2.82·8-s + (−7.24 − 4.18i)10-s + (−0.878 + 1.52i)11-s + 18.7i·13-s + (−2.00 − 3.46i)16-s + (20.3 + 11.7i)17-s + (−19.9 + 11.5i)19-s + 11.8i·20-s + 2.48·22-s + (9.36 + 16.2i)23-s + (4.98 − 8.63i)25-s + (22.9 − 13.2i)26-s + ⋯ |
| L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (1.02 − 0.591i)5-s + 0.353·8-s + (−0.724 − 0.418i)10-s + (−0.0798 + 0.138i)11-s + 1.44i·13-s + (−0.125 − 0.216i)16-s + (1.19 + 0.690i)17-s + (−1.05 + 0.606i)19-s + 0.591i·20-s + 0.112·22-s + (0.407 + 0.705i)23-s + (0.199 − 0.345i)25-s + (0.883 − 0.510i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 - 0.444i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.587284962\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.587284962\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.707 + 1.22i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + (-5.12 + 2.95i)T + (12.5 - 21.6i)T^{2} \) |
| 11 | \( 1 + (0.878 - 1.52i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 - 18.7iT - 169T^{2} \) |
| 17 | \( 1 + (-20.3 - 11.7i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (19.9 - 11.5i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-9.36 - 16.2i)T + (-264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + 30T + 841T^{2} \) |
| 31 | \( 1 + (7.45 + 4.30i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (-35.4 - 61.4i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + 41.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 10.4T + 1.84e3T^{2} \) |
| 47 | \( 1 + (33.5 - 19.3i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (18.5 - 32.0i)T + (-1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (84.4 + 48.7i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-14.4 + 8.36i)T + (1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (30.4 - 52.8i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 - 110.T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-49.1 - 28.3i)T + (2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-34.9 - 60.4i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + 6.43iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (-36.4 + 21.0i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 - 51.7iT - 9.40e3T^{2} \) |
| 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
−9.750670089573245793399690461775, −9.452116927204578315815848243399, −8.542196976228047396618197471072, −7.66148462670864129895831120078, −6.46640203958892971100473053435, −5.61937587440355028132122202181, −4.56290672041748406329784216334, −3.53808101983755560309554913366, −2.04423729808501822801086194257, −1.39075300145093798996549810025,
0.60276316800708090908506125115, 2.22549953683339364032242070869, 3.28203970352737067231800353767, 4.86482105997887005887046534551, 5.73515898175766480078559266322, 6.33098478832578716008250703916, 7.37251044737047616814622825600, 8.062234639452090632878664287786, 9.135151760494832305214671612265, 9.804245807762838170763326342646
