| L(s) = 1 | + (−6.02 + 1.37i)2-s + (4.35 + 14.1i)3-s + (20.0 − 9.63i)4-s + (39.0 + 15.3i)5-s + (−45.6 − 79.0i)6-s + (34.4 + 19.8i)7-s + (−29.9 + 23.8i)8-s + (−113. + 77.1i)9-s + (−256. − 38.6i)10-s + (81.4 + 39.2i)11-s + (222. + 240. i)12-s + (−57.9 + 8.72i)13-s + (−234. − 72.4i)14-s + (−46.2 + 617. i)15-s + (−73.7 + 92.5i)16-s + (−195. − 497. i)17-s + ⋯ |
| L(s) = 1 | + (−1.50 + 0.343i)2-s + (0.483 + 1.56i)3-s + (1.25 − 0.602i)4-s + (1.56 + 0.612i)5-s + (−1.26 − 2.19i)6-s + (0.702 + 0.405i)7-s + (−0.468 + 0.373i)8-s + (−1.39 + 0.952i)9-s + (−2.56 − 0.386i)10-s + (0.672 + 0.324i)11-s + (1.54 + 1.66i)12-s + (−0.342 + 0.0516i)13-s + (−1.19 − 0.369i)14-s + (−0.205 + 2.74i)15-s + (−0.288 + 0.361i)16-s + (−0.676 − 1.72i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.617 - 0.786i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.617 - 0.786i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.485664 + 0.998730i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.485664 + 0.998730i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (790. + 1.67e3i)T \) |
| good | 2 | \( 1 + (6.02 - 1.37i)T + (14.4 - 6.94i)T^{2} \) |
| 3 | \( 1 + (-4.35 - 14.1i)T + (-66.9 + 45.6i)T^{2} \) |
| 5 | \( 1 + (-39.0 - 15.3i)T + (458. + 425. i)T^{2} \) |
| 7 | \( 1 + (-34.4 - 19.8i)T + (1.20e3 + 2.07e3i)T^{2} \) |
| 11 | \( 1 + (-81.4 - 39.2i)T + (9.12e3 + 1.14e4i)T^{2} \) |
| 13 | \( 1 + (57.9 - 8.72i)T + (2.72e4 - 8.41e3i)T^{2} \) |
| 17 | \( 1 + (195. + 497. i)T + (-6.12e4 + 5.68e4i)T^{2} \) |
| 19 | \( 1 + (-141. + 208. i)T + (-4.76e4 - 1.21e5i)T^{2} \) |
| 23 | \( 1 + (43.4 + 579. i)T + (-2.76e5 + 4.17e4i)T^{2} \) |
| 29 | \( 1 + (-129. + 419. i)T + (-5.84e5 - 3.98e5i)T^{2} \) |
| 31 | \( 1 + (1.13e3 - 1.05e3i)T + (6.90e4 - 9.20e5i)T^{2} \) |
| 37 | \( 1 + (115. - 66.5i)T + (9.37e5 - 1.62e6i)T^{2} \) |
| 41 | \( 1 + (-266. - 1.16e3i)T + (-2.54e6 + 1.22e6i)T^{2} \) |
| 47 | \( 1 + (-3.04e3 + 1.46e3i)T + (3.04e6 - 3.81e6i)T^{2} \) |
| 53 | \( 1 + (3.33e3 + 502. i)T + (7.53e6 + 2.32e6i)T^{2} \) |
| 59 | \( 1 + (-1.25e3 + 1.57e3i)T + (-2.69e6 - 1.18e7i)T^{2} \) |
| 61 | \( 1 + (275. - 296. i)T + (-1.03e6 - 1.38e7i)T^{2} \) |
| 67 | \( 1 + (-685. - 467. i)T + (7.36e6 + 1.87e7i)T^{2} \) |
| 71 | \( 1 + (-473. - 35.4i)T + (2.51e7 + 3.78e6i)T^{2} \) |
| 73 | \( 1 + (-979. - 6.50e3i)T + (-2.71e7 + 8.37e6i)T^{2} \) |
| 79 | \( 1 + (-2.41e3 + 4.18e3i)T + (-1.94e7 - 3.37e7i)T^{2} \) |
| 83 | \( 1 + (1.51e3 - 468. i)T + (3.92e7 - 2.67e7i)T^{2} \) |
| 89 | \( 1 + (-2.33e3 - 7.55e3i)T + (-5.18e7 + 3.53e7i)T^{2} \) |
| 97 | \( 1 + (3.96e3 + 1.90e3i)T + (5.51e7 + 6.92e7i)T^{2} \) |
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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
−15.82442209706880129307933301553, −14.73447687200422365783770719892, −13.92178184362524689612015405345, −11.21353880575866602748761400898, −10.20886809523581084716113831523, −9.417461967451533698692431158443, −8.832864627176928414356137220517, −6.87771813403161048304085725706, −5.01596805066164599095140179143, −2.35064788646462213120761600009,
1.31567027989520850789539660937, 1.90837109930236658517262093005, 6.06206387042262860591406534520, 7.53914823673825706904444630562, 8.579182720858584156310782368181, 9.513633150965870680965600440888, 10.96291879999850057501793250383, 12.45870622336540646577677177112, 13.52401343134998394318004416405, 14.38256928485743343207017338590
