| L(s) = 1 | + (3.83 + 0.335i)2-s + (−3.82 + 3.51i)3-s + (6.74 + 1.18i)4-s + (−10.6 + 3.45i)5-s + (−15.8 + 12.2i)6-s + (−8.65 − 6.06i)7-s + (−4.30 − 1.15i)8-s + (2.28 − 26.9i)9-s + (−41.9 + 9.70i)10-s + (−18.9 + 51.9i)11-s + (−29.9 + 19.1i)12-s + (−0.436 − 4.98i)13-s + (−31.1 − 26.1i)14-s + (28.5 − 50.6i)15-s + (−67.5 − 24.5i)16-s + (3.33 − 0.892i)17-s + ⋯ |
| L(s) = 1 | + (1.35 + 0.118i)2-s + (−0.736 + 0.676i)3-s + (0.842 + 0.148i)4-s + (−0.950 + 0.309i)5-s + (−1.07 + 0.830i)6-s + (−0.467 − 0.327i)7-s + (−0.190 − 0.0509i)8-s + (0.0847 − 0.996i)9-s + (−1.32 + 0.307i)10-s + (−0.518 + 1.42i)11-s + (−0.720 + 0.460i)12-s + (−0.00930 − 0.106i)13-s + (−0.595 − 0.499i)14-s + (0.490 − 0.871i)15-s + (−1.05 − 0.384i)16-s + (0.0475 − 0.0127i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0462i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0462i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.0179922 + 0.778286i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0179922 + 0.778286i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (3.82 - 3.51i)T \) |
| 5 | \( 1 + (10.6 - 3.45i)T \) |
| good | 2 | \( 1 + (-3.83 - 0.335i)T + (7.87 + 1.38i)T^{2} \) |
| 7 | \( 1 + (8.65 + 6.06i)T + (117. + 322. i)T^{2} \) |
| 11 | \( 1 + (18.9 - 51.9i)T + (-1.01e3 - 855. i)T^{2} \) |
| 13 | \( 1 + (0.436 + 4.98i)T + (-2.16e3 + 381. i)T^{2} \) |
| 17 | \( 1 + (-3.33 + 0.892i)T + (4.25e3 - 2.45e3i)T^{2} \) |
| 19 | \( 1 + (26.0 - 15.0i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-49.6 - 70.8i)T + (-4.16e3 + 1.14e4i)T^{2} \) |
| 29 | \( 1 + (143. - 120. i)T + (4.23e3 - 2.40e4i)T^{2} \) |
| 31 | \( 1 + (2.89 - 16.4i)T + (-2.79e4 - 1.01e4i)T^{2} \) |
| 37 | \( 1 + (-103. - 387. i)T + (-4.38e4 + 2.53e4i)T^{2} \) |
| 41 | \( 1 + (-243. + 290. i)T + (-1.19e4 - 6.78e4i)T^{2} \) |
| 43 | \( 1 + (-45.4 - 97.5i)T + (-5.11e4 + 6.09e4i)T^{2} \) |
| 47 | \( 1 + (24.7 - 35.2i)T + (-3.55e4 - 9.75e4i)T^{2} \) |
| 53 | \( 1 + (256. - 256. i)T - 1.48e5iT^{2} \) |
| 59 | \( 1 + (192. - 70.2i)T + (1.57e5 - 1.32e5i)T^{2} \) |
| 61 | \( 1 + (136. + 776. i)T + (-2.13e5 + 7.76e4i)T^{2} \) |
| 67 | \( 1 + (816. - 71.3i)T + (2.96e5 - 5.22e4i)T^{2} \) |
| 71 | \( 1 + (-727. - 419. i)T + (1.78e5 + 3.09e5i)T^{2} \) |
| 73 | \( 1 + (28.5 - 106. i)T + (-3.36e5 - 1.94e5i)T^{2} \) |
| 79 | \( 1 + (603. + 719. i)T + (-8.56e4 + 4.85e5i)T^{2} \) |
| 83 | \( 1 + (-41.7 + 476. i)T + (-5.63e5 - 9.92e4i)T^{2} \) |
| 89 | \( 1 + (-381. - 661. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (1.09e3 - 509. i)T + (5.86e5 - 6.99e5i)T^{2} \) |
| 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
−12.99586434174867907036192804882, −12.39723799595529312462638788387, −11.48847543987927461722450742398, −10.47699792819903134579332026693, −9.355110887691349760276660355232, −7.43404666802345109155789782588, −6.47474929822196092232761270237, −5.11618300221133379141637968895, −4.27045558222265606332978892484, −3.23996814360986952926493471863,
0.26552686282441608855390545671, 2.83540242000287331422669021765, 4.24674955878211921449965557336, 5.49487117917718791494783018856, 6.29972873868769624813670566081, 7.69319899367226818940862350682, 8.892827182246705128621737184491, 10.97729869113103247610825145958, 11.45871539040162966329176958979, 12.57391762884394210400501001207
