| 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
−13.34748405693205485814371073669, −12.79850414274478452779951107601, −10.78552120944419486873814522616, −9.960730028206075123908905673869, −9.346062703853957274660208479416, −7.57703415123154064549529455371, −6.82020291181975930383957594767, −5.25897824143185551992376177280, −3.12080788026665186586440894115, −2.21248708102104367661823566659,
1.55021863977519299226219169787, 3.11896245926405056029727988085, 5.03125961629524131731693717162, 6.12026321225079203722752451219, 7.962527700964403888113425778082, 8.909442773282598595259034593764, 9.639683606714002223451308916117, 10.69936597227695810955948417351, 12.80280994484472076058428458086, 13.03460627862015527456946376508
