L(s) = 1 | + (0.998 − 2.40i)3-s + (−17.4 + 7.20i)5-s + (4.37 − 4.37i)7-s + (14.2 + 14.2i)9-s + (−11.7 − 28.4i)11-s + (12.9 + 5.35i)13-s + 49.1i·15-s − 72.9i·17-s + (143. + 59.2i)19-s + (−6.17 − 14.8i)21-s + (83.6 + 83.6i)23-s + (162. − 162. i)25-s + (113. − 47.1i)27-s + (39.6 − 95.7i)29-s − 29.0·31-s + ⋯ |
L(s) = 1 | + (0.192 − 0.463i)3-s + (−1.55 + 0.644i)5-s + (0.236 − 0.236i)7-s + (0.528 + 0.528i)9-s + (−0.323 − 0.780i)11-s + (0.275 + 0.114i)13-s + 0.845i·15-s − 1.04i·17-s + (1.72 + 0.715i)19-s + (−0.0641 − 0.154i)21-s + (0.758 + 0.758i)23-s + (1.29 − 1.29i)25-s + (0.810 − 0.335i)27-s + (0.253 − 0.613i)29-s − 0.168·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.937 + 0.349i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.937 + 0.349i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.537100330\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.537100330\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( 1 + (-0.998 + 2.40i)T + (-19.0 - 19.0i)T^{2} \) |
| 5 | \( 1 + (17.4 - 7.20i)T + (88.3 - 88.3i)T^{2} \) |
| 7 | \( 1 + (-4.37 + 4.37i)T - 343iT^{2} \) |
| 11 | \( 1 + (11.7 + 28.4i)T + (-941. + 941. i)T^{2} \) |
| 13 | \( 1 + (-12.9 - 5.35i)T + (1.55e3 + 1.55e3i)T^{2} \) |
| 17 | \( 1 + 72.9iT - 4.91e3T^{2} \) |
| 19 | \( 1 + (-143. - 59.2i)T + (4.85e3 + 4.85e3i)T^{2} \) |
| 23 | \( 1 + (-83.6 - 83.6i)T + 1.21e4iT^{2} \) |
| 29 | \( 1 + (-39.6 + 95.7i)T + (-1.72e4 - 1.72e4i)T^{2} \) |
| 31 | \( 1 + 29.0T + 2.97e4T^{2} \) |
| 37 | \( 1 + (-267. + 110. i)T + (3.58e4 - 3.58e4i)T^{2} \) |
| 41 | \( 1 + (124. + 124. i)T + 6.89e4iT^{2} \) |
| 43 | \( 1 + (-27.0 - 65.2i)T + (-5.62e4 + 5.62e4i)T^{2} \) |
| 47 | \( 1 + 282. iT - 1.03e5T^{2} \) |
| 53 | \( 1 + (-51.4 - 124. i)T + (-1.05e5 + 1.05e5i)T^{2} \) |
| 59 | \( 1 + (-222. + 92.1i)T + (1.45e5 - 1.45e5i)T^{2} \) |
| 61 | \( 1 + (-226. + 547. i)T + (-1.60e5 - 1.60e5i)T^{2} \) |
| 67 | \( 1 + (356. - 859. i)T + (-2.12e5 - 2.12e5i)T^{2} \) |
| 71 | \( 1 + (-690. + 690. i)T - 3.57e5iT^{2} \) |
| 73 | \( 1 + (-223. - 223. i)T + 3.89e5iT^{2} \) |
| 79 | \( 1 - 698. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + (-915. - 379. i)T + (4.04e5 + 4.04e5i)T^{2} \) |
| 89 | \( 1 + (163. - 163. i)T - 7.04e5iT^{2} \) |
| 97 | \( 1 + 839.T + 9.12e5T^{2} \) |
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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
−11.44362355924631054313023427676, −10.90602925963224987330178934396, −9.628182261049524459918981209619, −8.161112749648797263996474885609, −7.62392838604587139535227499819, −6.93304726063663081413941171953, −5.27150008433612221585087969897, −3.92248753332105263889963180984, −2.89397141311623013649124292036, −0.866834755161463861021145505250,
0.986156100639835726668293429252, 3.22954593704427702564411826582, 4.28788312833203178046019356550, 5.07243175338820143876208418459, 6.87308958481365311776020436938, 7.81458415007313603226850166556, 8.683224998489622152251117529963, 9.598977112861431129522825792743, 10.75553229109396002354882637374, 11.73216249499852687091546864716