| L(s) = 1 | + (1.23 + 3.54i)2-s + (−7.88 + 6.28i)4-s + (0.825 − 1.71i)5-s + (−2.97 + 3.72i)7-s + (−19.3 − 12.1i)8-s + (7.09 + 0.798i)10-s + (−8.99 + 5.65i)11-s + (−14.5 − 3.31i)13-s + (−16.8 − 5.91i)14-s + (10.0 − 44.1i)16-s + (1.89 + 1.89i)17-s + (−0.860 + 7.63i)19-s + (4.26 + 18.6i)20-s + (−31.1 − 24.8i)22-s + (12.8 − 6.21i)23-s + ⋯ |
| L(s) = 1 | + (0.619 + 1.77i)2-s + (−1.97 + 1.57i)4-s + (0.165 − 0.342i)5-s + (−0.424 + 0.532i)7-s + (−2.41 − 1.51i)8-s + (0.709 + 0.0798i)10-s + (−0.818 + 0.514i)11-s + (−1.11 − 0.254i)13-s + (−1.20 − 0.422i)14-s + (0.629 − 2.75i)16-s + (0.111 + 0.111i)17-s + (−0.0452 + 0.401i)19-s + (0.213 + 0.934i)20-s + (−1.41 − 1.13i)22-s + (0.560 − 0.270i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.443 + 0.896i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.443 + 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.574271 - 0.924880i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.574271 - 0.924880i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 29 | \( 1 + (-13.6 - 25.5i)T \) |
| good | 2 | \( 1 + (-1.23 - 3.54i)T + (-3.12 + 2.49i)T^{2} \) |
| 5 | \( 1 + (-0.825 + 1.71i)T + (-15.5 - 19.5i)T^{2} \) |
| 7 | \( 1 + (2.97 - 3.72i)T + (-10.9 - 47.7i)T^{2} \) |
| 11 | \( 1 + (8.99 - 5.65i)T + (52.4 - 109. i)T^{2} \) |
| 13 | \( 1 + (14.5 + 3.31i)T + (152. + 73.3i)T^{2} \) |
| 17 | \( 1 + (-1.89 - 1.89i)T + 289iT^{2} \) |
| 19 | \( 1 + (0.860 - 7.63i)T + (-351. - 80.3i)T^{2} \) |
| 23 | \( 1 + (-12.8 + 6.21i)T + (329. - 413. i)T^{2} \) |
| 31 | \( 1 + (5.75 + 16.4i)T + (-751. + 599. i)T^{2} \) |
| 37 | \( 1 + (-47.2 - 29.6i)T + (593. + 1.23e3i)T^{2} \) |
| 41 | \( 1 + (52.6 - 52.6i)T - 1.68e3iT^{2} \) |
| 43 | \( 1 + (57.3 + 20.0i)T + (1.44e3 + 1.15e3i)T^{2} \) |
| 47 | \( 1 + (-14.1 - 22.4i)T + (-958. + 1.99e3i)T^{2} \) |
| 53 | \( 1 + (-7.05 - 3.39i)T + (1.75e3 + 2.19e3i)T^{2} \) |
| 59 | \( 1 - 16.5T + 3.48e3T^{2} \) |
| 61 | \( 1 + (-45.2 + 5.09i)T + (3.62e3 - 828. i)T^{2} \) |
| 67 | \( 1 + (9.62 - 2.19i)T + (4.04e3 - 1.94e3i)T^{2} \) |
| 71 | \( 1 + (70.8 + 16.1i)T + (4.54e3 + 2.18e3i)T^{2} \) |
| 73 | \( 1 + (-26.3 + 75.3i)T + (-4.16e3 - 3.32e3i)T^{2} \) |
| 79 | \( 1 + (34.8 - 55.4i)T + (-2.70e3 - 5.62e3i)T^{2} \) |
| 83 | \( 1 + (29.5 + 37.0i)T + (-1.53e3 + 6.71e3i)T^{2} \) |
| 89 | \( 1 + (8.20 + 23.4i)T + (-6.19e3 + 4.93e3i)T^{2} \) |
| 97 | \( 1 + (-77.3 - 8.72i)T + (9.17e3 + 2.09e3i)T^{2} \) |
| 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
−12.89785131995133564776254248001, −11.97532222977275940980606603541, −10.11448929474902234778483976475, −9.186770999472456722796610110493, −8.210139679175815600461153668845, −7.35484604590792142703810026944, −6.39833202085295085853670368988, −5.29402230969989613587429959935, −4.71312211274410255128017458083, −3.03184594651606586040467246315,
0.44402688051715337043425495504, 2.32991657865510172337993667582, 3.24232220191489512077728926497, 4.51132707594662109899153549766, 5.50429255393667395616470784237, 7.00279394729976314074261177474, 8.601186996237721769807095423660, 9.822132348100191783709047181758, 10.27215731949898598600491323286, 11.16499346581572382306900731024
