L(s) = 1 | + (0.155 + 0.0897i)2-s + (5.17 + 0.514i)3-s + (−3.98 − 6.90i)4-s + (−2.5 + 4.33i)5-s + (0.757 + 0.544i)6-s + (18.1 + 3.54i)7-s − 2.86i·8-s + (26.4 + 5.31i)9-s + (−0.777 + 0.448i)10-s + (42.8 − 24.7i)11-s + (−17.0 − 37.7i)12-s − 62.5i·13-s + (2.50 + 2.18i)14-s + (−15.1 + 21.1i)15-s + (−31.6 + 54.7i)16-s + (19.4 + 33.6i)17-s + ⋯ |
L(s) = 1 | + (0.0549 + 0.0317i)2-s + (0.995 + 0.0989i)3-s + (−0.497 − 0.862i)4-s + (−0.223 + 0.387i)5-s + (0.0515 + 0.0370i)6-s + (0.981 + 0.191i)7-s − 0.126i·8-s + (0.980 + 0.196i)9-s + (−0.0245 + 0.0141i)10-s + (1.17 − 0.677i)11-s + (−0.410 − 0.907i)12-s − 1.33i·13-s + (0.0478 + 0.0416i)14-s + (−0.260 + 0.363i)15-s + (−0.493 + 0.855i)16-s + (0.277 + 0.480i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.913 + 0.406i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.913 + 0.406i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.13710 - 0.453903i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.13710 - 0.453903i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-5.17 - 0.514i)T \) |
| 5 | \( 1 + (2.5 - 4.33i)T \) |
| 7 | \( 1 + (-18.1 - 3.54i)T \) |
good | 2 | \( 1 + (-0.155 - 0.0897i)T + (4 + 6.92i)T^{2} \) |
| 11 | \( 1 + (-42.8 + 24.7i)T + (665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + 62.5iT - 2.19e3T^{2} \) |
| 17 | \( 1 + (-19.4 - 33.6i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-14.1 - 8.18i)T + (3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (125. + 72.6i)T + (6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 - 246. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + (-128. + 74.0i)T + (1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (174. - 301. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + 429.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 73.5T + 7.95e4T^{2} \) |
| 47 | \( 1 + (124. - 214. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (263. - 152. i)T + (7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (336. + 582. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (279. + 161. i)T + (1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (103. + 179. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 - 717. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (-84.3 + 48.7i)T + (1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-450. + 779. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 - 90.9T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-328. + 568. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 - 1.46e3iT - 9.12e5T^{2} \) |
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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
−13.64220278024304702690311801906, −12.24617697289822251715660654398, −10.83613681842750276503973560278, −10.00503191496080437292090591325, −8.719435740445633512940736171406, −8.016331337146138028502222072348, −6.33882780639524779732745003835, −4.85620770377214408652164117466, −3.45026258385807959322490083848, −1.44095739361936607102003000227,
1.83445272968235678017430212006, 3.85166299193527752347953722473, 4.57562426235185469323027701335, 7.03468807619538435756618577898, 7.959603426402136027575634801169, 8.905500167193157656895184000582, 9.704708341738929064033560985662, 11.76248851961166384065124295438, 12.14127417869027157165849081982, 13.76450572304564095610160845599