| L(s) = 1 | + (7.22 − 6.05i)3-s + (72.9 + 26.5i)5-s + (−122. + 212. i)7-s + (−26.7 + 151. i)9-s + (−288. − 500. i)11-s + (570. + 478. i)13-s + (687. − 250. i)15-s + (76.7 + 435. i)17-s + (328. + 1.53e3i)19-s + (402. + 2.28e3i)21-s + (−260. + 94.8i)23-s + (2.21e3 + 1.85e3i)25-s + (1.87e3 + 3.24e3i)27-s + (867. − 4.91e3i)29-s + (334. − 579. i)31-s + ⋯ |
| L(s) = 1 | + (0.463 − 0.388i)3-s + (1.30 + 0.474i)5-s + (−0.947 + 1.64i)7-s + (−0.110 + 0.624i)9-s + (−0.719 − 1.24i)11-s + (0.936 + 0.785i)13-s + (0.788 − 0.287i)15-s + (0.0643 + 0.365i)17-s + (0.208 + 0.978i)19-s + (0.198 + 1.12i)21-s + (−0.102 + 0.0373i)23-s + (0.709 + 0.595i)25-s + (0.494 + 0.855i)27-s + (0.191 − 1.08i)29-s + (0.0625 − 0.108i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.454 - 0.890i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.454 - 0.890i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.83789 + 1.12490i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.83789 + 1.12490i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (-328. - 1.53e3i)T \) |
| good | 3 | \( 1 + (-7.22 + 6.05i)T + (42.1 - 239. i)T^{2} \) |
| 5 | \( 1 + (-72.9 - 26.5i)T + (2.39e3 + 2.00e3i)T^{2} \) |
| 7 | \( 1 + (122. - 212. i)T + (-8.40e3 - 1.45e4i)T^{2} \) |
| 11 | \( 1 + (288. + 500. i)T + (-8.05e4 + 1.39e5i)T^{2} \) |
| 13 | \( 1 + (-570. - 478. i)T + (6.44e4 + 3.65e5i)T^{2} \) |
| 17 | \( 1 + (-76.7 - 435. i)T + (-1.33e6 + 4.85e5i)T^{2} \) |
| 23 | \( 1 + (260. - 94.8i)T + (4.93e6 - 4.13e6i)T^{2} \) |
| 29 | \( 1 + (-867. + 4.91e3i)T + (-1.92e7 - 7.01e6i)T^{2} \) |
| 31 | \( 1 + (-334. + 579. i)T + (-1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 - 1.61e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + (3.52e3 - 2.95e3i)T + (2.01e7 - 1.14e8i)T^{2} \) |
| 43 | \( 1 + (-8.25e3 - 3.00e3i)T + (1.12e8 + 9.44e7i)T^{2} \) |
| 47 | \( 1 + (-3.66e3 + 2.07e4i)T + (-2.15e8 - 7.84e7i)T^{2} \) |
| 53 | \( 1 + (9.70e3 - 3.53e3i)T + (3.20e8 - 2.68e8i)T^{2} \) |
| 59 | \( 1 + (4.39e3 + 2.49e4i)T + (-6.71e8 + 2.44e8i)T^{2} \) |
| 61 | \( 1 + (5.14e3 - 1.87e3i)T + (6.46e8 - 5.42e8i)T^{2} \) |
| 67 | \( 1 + (6.31e3 - 3.58e4i)T + (-1.26e9 - 4.61e8i)T^{2} \) |
| 71 | \( 1 + (7.21e4 + 2.62e4i)T + (1.38e9 + 1.15e9i)T^{2} \) |
| 73 | \( 1 + (2.13e4 - 1.79e4i)T + (3.59e8 - 2.04e9i)T^{2} \) |
| 79 | \( 1 + (9.45e3 - 7.93e3i)T + (5.34e8 - 3.03e9i)T^{2} \) |
| 83 | \( 1 + (-3.53e4 + 6.11e4i)T + (-1.96e9 - 3.41e9i)T^{2} \) |
| 89 | \( 1 + (-4.80e4 - 4.03e4i)T + (9.69e8 + 5.49e9i)T^{2} \) |
| 97 | \( 1 + (1.39e4 + 7.91e4i)T + (-8.06e9 + 2.93e9i)T^{2} \) |
| 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
−13.55911169506915456280563227259, −13.03502385199860167982160989563, −11.54136621971619549447238089314, −10.22068222348120745846419564181, −9.122190271568600385558547094608, −8.180216596471570261781083198730, −6.15123787349339552378201283598, −5.80037833329413634320727860122, −2.99758861707372367909625448597, −2.04388865286562777107549328766,
0.910121265936099213957826039138, 3.00330862000791913402242726231, 4.52908313765261487650298714690, 6.16015629220786491882297642567, 7.40982441996665667031034095422, 9.159474943294463555718177280220, 9.880738851997134839500879136088, 10.65063593127784416228674940893, 12.76573610617043898357676504536, 13.29521201215270593767009239216
