L(s) = 1 | + (5.19 + 0.246i)3-s + (15.7 − 13.1i)5-s + (−8.81 + 3.20i)7-s + (26.8 + 2.56i)9-s + (−45.7 − 38.3i)11-s + (2.61 + 14.8i)13-s + (84.7 − 64.5i)15-s + (44.8 + 77.6i)17-s + (−3.79 + 6.57i)19-s + (−46.5 + 14.4i)21-s + (126. + 45.9i)23-s + (51.2 − 290. i)25-s + (138. + 19.9i)27-s + (−25.0 + 142. i)29-s + (−243. − 88.7i)31-s + ⋯ |
L(s) = 1 | + (0.998 + 0.0474i)3-s + (1.40 − 1.17i)5-s + (−0.476 + 0.173i)7-s + (0.995 + 0.0948i)9-s + (−1.25 − 1.05i)11-s + (0.0557 + 0.316i)13-s + (1.45 − 1.11i)15-s + (0.639 + 1.10i)17-s + (−0.0458 + 0.0794i)19-s + (−0.483 + 0.150i)21-s + (1.14 + 0.416i)23-s + (0.409 − 2.32i)25-s + (0.989 + 0.142i)27-s + (−0.160 + 0.910i)29-s + (−1.41 − 0.513i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.852 + 0.522i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.852 + 0.522i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.37365 - 0.669415i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.37365 - 0.669415i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-5.19 - 0.246i)T \) |
good | 5 | \( 1 + (-15.7 + 13.1i)T + (21.7 - 123. i)T^{2} \) |
| 7 | \( 1 + (8.81 - 3.20i)T + (262. - 220. i)T^{2} \) |
| 11 | \( 1 + (45.7 + 38.3i)T + (231. + 1.31e3i)T^{2} \) |
| 13 | \( 1 + (-2.61 - 14.8i)T + (-2.06e3 + 751. i)T^{2} \) |
| 17 | \( 1 + (-44.8 - 77.6i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (3.79 - 6.57i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-126. - 45.9i)T + (9.32e3 + 7.82e3i)T^{2} \) |
| 29 | \( 1 + (25.0 - 142. i)T + (-2.29e4 - 8.34e3i)T^{2} \) |
| 31 | \( 1 + (243. + 88.7i)T + (2.28e4 + 1.91e4i)T^{2} \) |
| 37 | \( 1 + (-113. - 195. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + (-32.6 - 185. i)T + (-6.47e4 + 2.35e4i)T^{2} \) |
| 43 | \( 1 + (45.0 + 37.7i)T + (1.38e4 + 7.82e4i)T^{2} \) |
| 47 | \( 1 + (298. - 108. i)T + (7.95e4 - 6.67e4i)T^{2} \) |
| 53 | \( 1 + 170.T + 1.48e5T^{2} \) |
| 59 | \( 1 + (82.5 - 69.3i)T + (3.56e4 - 2.02e5i)T^{2} \) |
| 61 | \( 1 + (324. - 118. i)T + (1.73e5 - 1.45e5i)T^{2} \) |
| 67 | \( 1 + (-1.02 - 5.80i)T + (-2.82e5 + 1.02e5i)T^{2} \) |
| 71 | \( 1 + (205. + 356. i)T + (-1.78e5 + 3.09e5i)T^{2} \) |
| 73 | \( 1 + (-602. + 1.04e3i)T + (-1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-19.8 + 112. i)T + (-4.63e5 - 1.68e5i)T^{2} \) |
| 83 | \( 1 + (245. - 1.39e3i)T + (-5.37e5 - 1.95e5i)T^{2} \) |
| 89 | \( 1 + (-683. + 1.18e3i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + (294. + 247. i)T + (1.58e5 + 8.98e5i)T^{2} \) |
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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
−13.03460627862015527456946376508, −12.80280994484472076058428458086, −10.69936597227695810955948417351, −9.639683606714002223451308916117, −8.909442773282598595259034593764, −7.962527700964403888113425778082, −6.12026321225079203722752451219, −5.03125961629524131731693717162, −3.11896245926405056029727988085, −1.55021863977519299226219169787,
2.21248708102104367661823566659, 3.12080788026665186586440894115, 5.25897824143185551992376177280, 6.82020291181975930383957594767, 7.57703415123154064549529455371, 9.346062703853957274660208479416, 9.960730028206075123908905673869, 10.78552120944419486873814522616, 12.79850414274478452779951107601, 13.34748405693205485814371073669