| L(s) = 1 | + (−1.41 + 2.44i)2-s + (−4.5 + 2.59i)3-s + (−3.99 − 6.92i)4-s + (−12.2 − 7.07i)5-s − 14.6i·6-s + (12.1 − 47.4i)7-s + 22.6·8-s + (13.5 − 23.3i)9-s + (34.6 − 20.0i)10-s + (−32.0 − 55.4i)11-s + (35.9 + 20.7i)12-s − 228. i·13-s + (98.9 + 96.9i)14-s + 73.5·15-s + (−32.0 + 55.4i)16-s + (−195. + 112. i)17-s + ⋯ |
| L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.5 + 0.288i)3-s + (−0.249 − 0.433i)4-s + (−0.490 − 0.283i)5-s − 0.408i·6-s + (0.248 − 0.968i)7-s + 0.353·8-s + (0.166 − 0.288i)9-s + (0.346 − 0.200i)10-s + (−0.264 − 0.458i)11-s + (0.249 + 0.144i)12-s − 1.35i·13-s + (0.505 + 0.494i)14-s + 0.326·15-s + (−0.125 + 0.216i)16-s + (−0.675 + 0.390i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.144 + 0.989i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.144 + 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.434685 - 0.375724i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.434685 - 0.375724i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.41 - 2.44i)T \) |
| 3 | \( 1 + (4.5 - 2.59i)T \) |
| 7 | \( 1 + (-12.1 + 47.4i)T \) |
| good | 5 | \( 1 + (12.2 + 7.07i)T + (312.5 + 541. i)T^{2} \) |
| 11 | \( 1 + (32.0 + 55.4i)T + (-7.32e3 + 1.26e4i)T^{2} \) |
| 13 | \( 1 + 228. iT - 2.85e4T^{2} \) |
| 17 | \( 1 + (195. - 112. i)T + (4.17e4 - 7.23e4i)T^{2} \) |
| 19 | \( 1 + (255. + 147. i)T + (6.51e4 + 1.12e5i)T^{2} \) |
| 23 | \( 1 + (354. - 614. i)T + (-1.39e5 - 2.42e5i)T^{2} \) |
| 29 | \( 1 - 740.T + 7.07e5T^{2} \) |
| 31 | \( 1 + (577. - 333. i)T + (4.61e5 - 7.99e5i)T^{2} \) |
| 37 | \( 1 + (-416. + 722. i)T + (-9.37e5 - 1.62e6i)T^{2} \) |
| 41 | \( 1 + 2.81e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 3.06e3T + 3.41e6T^{2} \) |
| 47 | \( 1 + (531. + 306. i)T + (2.43e6 + 4.22e6i)T^{2} \) |
| 53 | \( 1 + (-576. - 998. i)T + (-3.94e6 + 6.83e6i)T^{2} \) |
| 59 | \( 1 + (3.02e3 - 1.74e3i)T + (6.05e6 - 1.04e7i)T^{2} \) |
| 61 | \( 1 + (-1.96e3 - 1.13e3i)T + (6.92e6 + 1.19e7i)T^{2} \) |
| 67 | \( 1 + (-4.33e3 - 7.51e3i)T + (-1.00e7 + 1.74e7i)T^{2} \) |
| 71 | \( 1 + 353.T + 2.54e7T^{2} \) |
| 73 | \( 1 + (-3.52e3 + 2.03e3i)T + (1.41e7 - 2.45e7i)T^{2} \) |
| 79 | \( 1 + (3.23e3 - 5.60e3i)T + (-1.94e7 - 3.37e7i)T^{2} \) |
| 83 | \( 1 + 8.22e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + (1.34e4 + 7.76e3i)T + (3.13e7 + 5.43e7i)T^{2} \) |
| 97 | \( 1 - 1.55e3iT - 8.85e7T^{2} \) |
| 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
−15.42291075567251323983167011155, −13.97312949187078400325955798354, −12.72029639085235317854218287072, −11.07583073519812671282175951655, −10.22762298009413030304431728961, −8.513461754460515562636713109758, −7.36786516011411637452826745338, −5.72389449553646269190787369851, −4.15999634413400173504192926364, −0.44197560749307898514498397904,
2.18237728783842648973651337013, 4.51235265272485890990695384286, 6.51380775269341677071313114921, 8.095716347713537995701107822048, 9.443278270233122208696910349472, 10.99754582046008135712481247391, 11.82909976331571025632243752998, 12.75246868895794615501362838857, 14.33654687539134639787589153758, 15.65601737331845471113033254986
