L(s) = 1 | + (2.01 − 1.98i)2-s + (−1.20 − 2.90i)3-s + (0.125 − 7.99i)4-s + (−3.98 − 1.65i)5-s + (−8.17 − 3.46i)6-s + (22.4 + 22.4i)7-s + (−15.6 − 16.3i)8-s + (12.1 − 12.1i)9-s + (−11.3 + 4.58i)10-s + (−16.5 + 39.8i)11-s + (−23.3 + 9.25i)12-s + (17.9 − 7.42i)13-s + (89.6 + 0.701i)14-s + 13.5i·15-s + (−63.9 − 2.00i)16-s + 45.9i·17-s + ⋯ |
L(s) = 1 | + (0.712 − 0.701i)2-s + (−0.231 − 0.558i)3-s + (0.0156 − 0.999i)4-s + (−0.356 − 0.147i)5-s + (−0.556 − 0.235i)6-s + (1.20 + 1.20i)7-s + (−0.690 − 0.723i)8-s + (0.448 − 0.448i)9-s + (−0.357 + 0.144i)10-s + (−0.452 + 1.09i)11-s + (−0.561 + 0.222i)12-s + (0.382 − 0.158i)13-s + (1.71 + 0.0133i)14-s + 0.233i·15-s + (−0.999 − 0.0313i)16-s + 0.656i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.156 + 0.987i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.156 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.23195 - 1.05205i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.23195 - 1.05205i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-2.01 + 1.98i)T \) |
good | 3 | \( 1 + (1.20 + 2.90i)T + (-19.0 + 19.0i)T^{2} \) |
| 5 | \( 1 + (3.98 + 1.65i)T + (88.3 + 88.3i)T^{2} \) |
| 7 | \( 1 + (-22.4 - 22.4i)T + 343iT^{2} \) |
| 11 | \( 1 + (16.5 - 39.8i)T + (-941. - 941. i)T^{2} \) |
| 13 | \( 1 + (-17.9 + 7.42i)T + (1.55e3 - 1.55e3i)T^{2} \) |
| 17 | \( 1 - 45.9iT - 4.91e3T^{2} \) |
| 19 | \( 1 + (25.0 - 10.3i)T + (4.85e3 - 4.85e3i)T^{2} \) |
| 23 | \( 1 + (40.3 - 40.3i)T - 1.21e4iT^{2} \) |
| 29 | \( 1 + (88.6 + 214. i)T + (-1.72e4 + 1.72e4i)T^{2} \) |
| 31 | \( 1 - 260.T + 2.97e4T^{2} \) |
| 37 | \( 1 + (-70.4 - 29.1i)T + (3.58e4 + 3.58e4i)T^{2} \) |
| 41 | \( 1 + (251. - 251. i)T - 6.89e4iT^{2} \) |
| 43 | \( 1 + (95.7 - 231. i)T + (-5.62e4 - 5.62e4i)T^{2} \) |
| 47 | \( 1 + 15.5iT - 1.03e5T^{2} \) |
| 53 | \( 1 + (-171. + 414. i)T + (-1.05e5 - 1.05e5i)T^{2} \) |
| 59 | \( 1 + (-53.3 - 22.0i)T + (1.45e5 + 1.45e5i)T^{2} \) |
| 61 | \( 1 + (-297. - 718. i)T + (-1.60e5 + 1.60e5i)T^{2} \) |
| 67 | \( 1 + (377. + 911. i)T + (-2.12e5 + 2.12e5i)T^{2} \) |
| 71 | \( 1 + (359. + 359. i)T + 3.57e5iT^{2} \) |
| 73 | \( 1 + (-605. + 605. i)T - 3.89e5iT^{2} \) |
| 79 | \( 1 - 380. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + (-235. + 97.4i)T + (4.04e5 - 4.04e5i)T^{2} \) |
| 89 | \( 1 + (949. + 949. i)T + 7.04e5iT^{2} \) |
| 97 | \( 1 - 663.T + 9.12e5T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.40566003445412610917851532160, −14.99587072961142188907717574430, −13.30313845370715016445545406107, −12.17920031962068268027156896671, −11.60814098192033750681836450079, −9.913951094816552687895469744416, −8.091333415648116702502630553151, −6.08269221208307949748252674566, −4.51554204127214950483426253881, −1.91182618904851289325809070503,
3.94903825119567701968918598362, 5.18393732027127625281554319255, 7.21436759672264583671127352423, 8.329013265857765032724034282943, 10.64177705935803586366460170137, 11.51072746603930541206885148610, 13.40708704045695023870476475901, 14.14731020684461459331792316653, 15.49352500117496073851212797417, 16.40326653535827299940483478152