| L(s) = 1 | + (3.21 + 6.67i)2-s + (−3.25 + 6.75i)3-s + (−24.2 + 30.3i)4-s + (40.7 + 9.29i)5-s − 55.5·6-s − 88.1i·7-s + (−165. − 37.7i)8-s + (15.4 + 19.3i)9-s + (68.8 + 301. i)10-s + (−22.9 − 28.8i)11-s + (−126. − 262. i)12-s + (−8.88 + 38.9i)13-s + (588. − 283. i)14-s + (−195. + 244. i)15-s + (−140. − 617. i)16-s + (−5.67 − 24.8i)17-s + ⋯ |
| L(s) = 1 | + (0.803 + 1.66i)2-s + (−0.361 + 0.750i)3-s + (−1.51 + 1.89i)4-s + (1.62 + 0.371i)5-s − 1.54·6-s − 1.79i·7-s + (−2.58 − 0.589i)8-s + (0.190 + 0.239i)9-s + (0.688 + 3.01i)10-s + (−0.190 − 0.238i)11-s + (−0.878 − 1.82i)12-s + (−0.0525 + 0.230i)13-s + (3.00 − 1.44i)14-s + (−0.867 + 1.08i)15-s + (−0.550 − 2.41i)16-s + (−0.0196 − 0.0859i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.971 - 0.235i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.971 - 0.235i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.261668 + 2.18820i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.261668 + 2.18820i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-1.77e3 + 528. i)T \) |
| good | 2 | \( 1 + (-3.21 - 6.67i)T + (-9.97 + 12.5i)T^{2} \) |
| 3 | \( 1 + (3.25 - 6.75i)T + (-50.5 - 63.3i)T^{2} \) |
| 5 | \( 1 + (-40.7 - 9.29i)T + (563. + 271. i)T^{2} \) |
| 7 | \( 1 + 88.1iT - 2.40e3T^{2} \) |
| 11 | \( 1 + (22.9 + 28.8i)T + (-3.25e3 + 1.42e4i)T^{2} \) |
| 13 | \( 1 + (8.88 - 38.9i)T + (-2.57e4 - 1.23e4i)T^{2} \) |
| 17 | \( 1 + (5.67 + 24.8i)T + (-7.52e4 + 3.62e4i)T^{2} \) |
| 19 | \( 1 + (-351. - 280. i)T + (2.89e4 + 1.27e5i)T^{2} \) |
| 23 | \( 1 + (432. + 542. i)T + (-6.22e4 + 2.72e5i)T^{2} \) |
| 29 | \( 1 + (79.2 + 164. i)T + (-4.40e5 + 5.52e5i)T^{2} \) |
| 31 | \( 1 + (-62.5 + 30.1i)T + (5.75e5 - 7.22e5i)T^{2} \) |
| 37 | \( 1 + 545. iT - 1.87e6T^{2} \) |
| 41 | \( 1 + (-314. + 151. i)T + (1.76e6 - 2.20e6i)T^{2} \) |
| 47 | \( 1 + (-14.3 + 18.0i)T + (-1.08e6 - 4.75e6i)T^{2} \) |
| 53 | \( 1 + (6.08 + 26.6i)T + (-7.10e6 + 3.42e6i)T^{2} \) |
| 59 | \( 1 + (715. + 3.13e3i)T + (-1.09e7 + 5.25e6i)T^{2} \) |
| 61 | \( 1 + (740. - 1.53e3i)T + (-8.63e6 - 1.08e7i)T^{2} \) |
| 67 | \( 1 + (-3.35e3 + 4.21e3i)T + (-4.48e6 - 1.96e7i)T^{2} \) |
| 71 | \( 1 + (-3.40e3 - 2.71e3i)T + (5.65e6 + 2.47e7i)T^{2} \) |
| 73 | \( 1 + (-2.42e3 - 553. i)T + (2.55e7 + 1.23e7i)T^{2} \) |
| 79 | \( 1 + 7.49e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + (-483. - 232. i)T + (2.95e7 + 3.71e7i)T^{2} \) |
| 89 | \( 1 + (1.89e3 - 3.92e3i)T + (-3.91e7 - 4.90e7i)T^{2} \) |
| 97 | \( 1 + (5.81e3 + 7.29e3i)T + (-1.96e7 + 8.63e7i)T^{2} \) |
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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
−15.94076483006107927782748052610, −14.24079348009542467279925883464, −13.98420925237588495848821339063, −13.01043792948604606958003969085, −10.58441355026351297019396667352, −9.661888778542086919961002444601, −7.66090781555259461745794579059, −6.47961150565369616478533416794, −5.31411505533317385355977165354, −4.03222431742991465807965107364,
1.49950618493217414211508341522, 2.58384614928900371199555841956, 5.27088438749203027337970389028, 5.97227763482313665292080345996, 9.182756690718372432829259675052, 9.835841088287656564359257955572, 11.53174057077064963913947425328, 12.41697988989284546414558747855, 13.06261708050905070013090914944, 14.00944347743041741444244459656
