| L(s) = 1 | + (−14.5 − 5.78i)2-s + (45.9 + 5.00i)3-s + (130. + 123. i)4-s + (99.9 − 117. i)5-s + (−638. − 338. i)6-s + (−238. − 143. i)7-s + (−760. − 1.64e3i)8-s + (1.37e3 + 303. i)9-s + (−2.13e3 + 1.12e3i)10-s + (1.15e3 + 62.7i)11-s + (5.38e3 + 6.34e3i)12-s + (−651. − 2.96e3i)13-s + (2.63e3 + 3.46e3i)14-s + (5.18e3 − 4.91e3i)15-s + (907. + 1.67e4i)16-s + (−3.39e3 + 2.04e3i)17-s + ⋯ |
| L(s) = 1 | + (−1.81 − 0.722i)2-s + (1.70 + 0.185i)3-s + (2.04 + 1.93i)4-s + (0.799 − 0.941i)5-s + (−2.95 − 1.56i)6-s + (−0.696 − 0.419i)7-s + (−1.48 − 3.20i)8-s + (1.88 + 0.415i)9-s + (−2.13 + 1.12i)10-s + (0.869 + 0.0471i)11-s + (3.11 + 3.67i)12-s + (−0.296 − 1.34i)13-s + (0.960 + 1.26i)14-s + (1.53 − 1.45i)15-s + (0.221 + 4.08i)16-s + (−0.691 + 0.416i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0482 + 0.998i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.0482 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.05126 - 1.10324i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.05126 - 1.10324i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 59 | \( 1 + (-1.98e5 + 5.25e4i)T \) |
| good | 2 | \( 1 + (14.5 + 5.78i)T + (46.4 + 44.0i)T^{2} \) |
| 3 | \( 1 + (-45.9 - 5.00i)T + (711. + 156. i)T^{2} \) |
| 5 | \( 1 + (-99.9 + 117. i)T + (-2.52e3 - 1.54e4i)T^{2} \) |
| 7 | \( 1 + (238. + 143. i)T + (5.51e4 + 1.03e5i)T^{2} \) |
| 11 | \( 1 + (-1.15e3 - 62.7i)T + (1.76e6 + 1.91e5i)T^{2} \) |
| 13 | \( 1 + (651. + 2.96e3i)T + (-4.38e6 + 2.02e6i)T^{2} \) |
| 17 | \( 1 + (3.39e3 - 2.04e3i)T + (1.13e7 - 2.13e7i)T^{2} \) |
| 19 | \( 1 + (9.18 - 33.0i)T + (-4.03e7 - 2.42e7i)T^{2} \) |
| 23 | \( 1 + (-1.12e4 + 7.66e3i)T + (5.47e7 - 1.37e8i)T^{2} \) |
| 29 | \( 1 + (-5.99e3 - 1.50e4i)T + (-4.31e8 + 4.09e8i)T^{2} \) |
| 31 | \( 1 + (1.20e4 - 3.34e3i)T + (7.60e8 - 4.57e8i)T^{2} \) |
| 37 | \( 1 + (-2.98e4 + 6.44e4i)T + (-1.66e9 - 1.95e9i)T^{2} \) |
| 41 | \( 1 + (-1.17e4 + 1.73e4i)T + (-1.75e9 - 4.41e9i)T^{2} \) |
| 43 | \( 1 + (8.76e4 - 4.75e3i)T + (6.28e9 - 6.83e8i)T^{2} \) |
| 47 | \( 1 + (-2.27e4 + 1.93e4i)T + (1.74e9 - 1.06e10i)T^{2} \) |
| 53 | \( 1 + (6.20e4 - 1.17e5i)T + (-1.24e10 - 1.83e10i)T^{2} \) |
| 61 | \( 1 + (-3.35e5 - 1.33e5i)T + (3.74e10 + 3.54e10i)T^{2} \) |
| 67 | \( 1 + (8.86e4 + 1.91e5i)T + (-5.85e10 + 6.89e10i)T^{2} \) |
| 71 | \( 1 + (-1.32e5 - 1.55e5i)T + (-2.07e10 + 1.26e11i)T^{2} \) |
| 73 | \( 1 + (-2.97e4 - 3.91e4i)T + (-4.04e10 + 1.45e11i)T^{2} \) |
| 79 | \( 1 + (-5.09e5 + 5.53e4i)T + (2.37e11 - 5.22e10i)T^{2} \) |
| 83 | \( 1 + (-4.13e4 - 1.22e5i)T + (-2.60e11 + 1.97e11i)T^{2} \) |
| 89 | \( 1 + (1.12e5 - 4.48e4i)T + (3.60e11 - 3.41e11i)T^{2} \) |
| 97 | \( 1 + (1.01e6 - 1.33e6i)T + (-2.22e11 - 8.02e11i)T^{2} \) |
| 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
−13.17842045126496153995780461923, −12.63422407433817777675024792239, −10.60436104101054736292192896035, −9.651907979434952911967441456429, −9.037705862847888199579775450092, −8.268880168201946092844095818086, −6.96567573540581604665915134954, −3.56328530672497930613245054611, −2.30505372631906119674211350012, −0.972033458043886487736252897757,
1.75296324935313023634711915907, 2.74847386742283004333190029810, 6.51606669121723199816699419985, 7.01984644826513738921377128293, 8.492510089821092042047482959177, 9.480728369614491140511067608330, 9.714438142429367538915748396396, 11.40319686717266526557860917335, 13.66899702158180470534919850970, 14.57191681257100252040687120642
