| L(s) = 1 | + (10.2 + 8.20i)2-s + (−4.13 − 27.4i)3-s + (24.2 + 106. i)4-s + (86.4 + 126. i)5-s + (182. − 316. i)6-s + (220. − 127. i)7-s + (−257. + 534. i)8-s + (−40.0 + 12.3i)9-s + (−150. + 2.01e3i)10-s + (−297. + 1.30e3i)11-s + (2.82e3 − 1.10e3i)12-s + (86.6 + 1.15e3i)13-s + (3.30e3 + 498. i)14-s + (3.12e3 − 2.89e3i)15-s + (−743. + 358. i)16-s + (345. + 235. i)17-s + ⋯ |
| L(s) = 1 | + (1.28 + 1.02i)2-s + (−0.153 − 1.01i)3-s + (0.379 + 1.66i)4-s + (0.691 + 1.01i)5-s + (0.845 − 1.46i)6-s + (0.641 − 0.370i)7-s + (−0.502 + 1.04i)8-s + (−0.0550 + 0.0169i)9-s + (−0.150 + 2.01i)10-s + (−0.223 + 0.980i)11-s + (1.63 − 0.640i)12-s + (0.0394 + 0.526i)13-s + (1.20 + 0.181i)14-s + (0.925 − 0.859i)15-s + (−0.181 + 0.0874i)16-s + (0.0702 + 0.0478i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.441 - 0.897i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.441 - 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(3.15392 + 1.96250i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.15392 + 1.96250i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-7.91e4 - 7.79e3i)T \) |
| good | 2 | \( 1 + (-10.2 - 8.20i)T + (14.2 + 62.3i)T^{2} \) |
| 3 | \( 1 + (4.13 + 27.4i)T + (-696. + 214. i)T^{2} \) |
| 5 | \( 1 + (-86.4 - 126. i)T + (-5.70e3 + 1.45e4i)T^{2} \) |
| 7 | \( 1 + (-220. + 127. i)T + (5.88e4 - 1.01e5i)T^{2} \) |
| 11 | \( 1 + (297. - 1.30e3i)T + (-1.59e6 - 7.68e5i)T^{2} \) |
| 13 | \( 1 + (-86.6 - 1.15e3i)T + (-4.77e6 + 7.19e5i)T^{2} \) |
| 17 | \( 1 + (-345. - 235. i)T + (8.81e6 + 2.24e7i)T^{2} \) |
| 19 | \( 1 + (122. - 395. i)T + (-3.88e7 - 2.65e7i)T^{2} \) |
| 23 | \( 1 + (1.14e4 + 1.06e4i)T + (1.10e7 + 1.47e8i)T^{2} \) |
| 29 | \( 1 + (-4.02e3 + 2.67e4i)T + (-5.68e8 - 1.75e8i)T^{2} \) |
| 31 | \( 1 + (8.49e3 + 2.16e4i)T + (-6.50e8 + 6.03e8i)T^{2} \) |
| 37 | \( 1 + (7.90e4 + 4.56e4i)T + (1.28e9 + 2.22e9i)T^{2} \) |
| 41 | \( 1 + (4.29e4 - 5.37e4i)T + (-1.05e9 - 4.63e9i)T^{2} \) |
| 47 | \( 1 + (1.39e4 + 6.12e4i)T + (-9.71e9 + 4.67e9i)T^{2} \) |
| 53 | \( 1 + (1.66e4 - 2.22e5i)T + (-2.19e10 - 3.30e9i)T^{2} \) |
| 59 | \( 1 + (-2.32e5 + 1.11e5i)T + (2.62e10 - 3.29e10i)T^{2} \) |
| 61 | \( 1 + (6.65e4 + 2.61e4i)T + (3.77e10 + 3.50e10i)T^{2} \) |
| 67 | \( 1 + (1.48e5 + 4.58e4i)T + (7.47e10 + 5.09e10i)T^{2} \) |
| 71 | \( 1 + (-3.95e5 - 4.26e5i)T + (-9.57e9 + 1.27e11i)T^{2} \) |
| 73 | \( 1 + (5.14e5 - 3.85e4i)T + (1.49e11 - 2.25e10i)T^{2} \) |
| 79 | \( 1 + (2.31e5 + 4.01e5i)T + (-1.21e11 + 2.10e11i)T^{2} \) |
| 83 | \( 1 + (-4.98e5 + 7.51e4i)T + (3.12e11 - 9.63e10i)T^{2} \) |
| 89 | \( 1 + (1.86e5 + 1.23e6i)T + (-4.74e11 + 1.46e11i)T^{2} \) |
| 97 | \( 1 + (2.65e5 - 1.16e6i)T + (-7.50e11 - 3.61e11i)T^{2} \) |
| 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
−14.51430574733426608303145173431, −13.95815814444004147449063986491, −12.93923330172337455479670976857, −11.89518018586079773042451082893, −10.20124656969372568460815442481, −7.73721387203633387974551754661, −6.91407334362316810080441263521, −6.01801420631504307314782456819, −4.32513685157142267079797872684, −2.13050351355204311668115447512,
1.57371596443945463843628989205, 3.51790948200780681011100146077, 5.02532748645442495511134983539, 5.50020656585245499152737009251, 8.646262632813778480713644859612, 10.05345920707468131523311848517, 11.00543332792992062797683667410, 12.19053333264715330189670393121, 13.26100144164118437884274185741, 14.17150116507556415119588780023
