| L(s) = 1 | + (−4.33 − 2.50i)2-s + (−5.18 − 0.400i)3-s + (8.54 + 14.8i)4-s + 18.5·5-s + (21.4 + 14.7i)6-s + (−17.9 + 4.62i)7-s − 45.5i·8-s + (26.6 + 4.14i)9-s + (−80.5 − 46.5i)10-s − 35.1i·11-s + (−38.3 − 80.1i)12-s + (3.94 + 2.27i)13-s + (89.3 + 24.8i)14-s + (−96.1 − 7.43i)15-s + (−45.7 + 79.2i)16-s + (20.7 − 36.0i)17-s + ⋯ |
| L(s) = 1 | + (−1.53 − 0.885i)2-s + (−0.997 − 0.0770i)3-s + (1.06 + 1.85i)4-s + 1.66·5-s + (1.46 + 1.00i)6-s + (−0.968 + 0.249i)7-s − 2.01i·8-s + (0.988 + 0.153i)9-s + (−2.54 − 1.47i)10-s − 0.963i·11-s + (−0.922 − 1.92i)12-s + (0.0841 + 0.0485i)13-s + (1.70 + 0.474i)14-s + (−1.65 − 0.128i)15-s + (−0.715 + 1.23i)16-s + (0.296 − 0.513i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.123 + 0.992i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.123 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.380843 - 0.431235i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.380843 - 0.431235i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (5.18 + 0.400i)T \) |
| 7 | \( 1 + (17.9 - 4.62i)T \) |
| good | 2 | \( 1 + (4.33 + 2.50i)T + (4 + 6.92i)T^{2} \) |
| 5 | \( 1 - 18.5T + 125T^{2} \) |
| 11 | \( 1 + 35.1iT - 1.33e3T^{2} \) |
| 13 | \( 1 + (-3.94 - 2.27i)T + (1.09e3 + 1.90e3i)T^{2} \) |
| 17 | \( 1 + (-20.7 + 36.0i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-64.8 + 37.4i)T + (3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + 152. iT - 1.21e4T^{2} \) |
| 29 | \( 1 + (-42.5 + 24.5i)T + (1.21e4 - 2.11e4i)T^{2} \) |
| 31 | \( 1 + (-87.6 + 50.6i)T + (1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-2.86 - 4.96i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + (6.35 - 10.9i)T + (-3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (-69.1 - 119. i)T + (-3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + (-269. + 466. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-105. - 60.8i)T + (7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-156. - 270. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (436. + 252. i)T + (1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-327. - 567. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 - 891. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (315. + 182. i)T + (1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (242. - 420. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + (248. + 430. i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + (685. + 1.18e3i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + (205. - 118. i)T + (4.56e5 - 7.90e5i)T^{2} \) |
| 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
−13.70081714409055130364739302737, −12.69424804807184885394989102824, −11.57692265371805655546681910626, −10.39772096394755400406753213188, −9.817806775633618568205656719559, −8.824126064846330189947672896145, −6.86946056938987019207891263960, −5.74576480776472584924248912891, −2.63914725127114981047002246391, −0.819734121032525316003192523731,
1.41755862372648913126549659428, 5.54599153654557112193986820087, 6.34831042451556053498137944915, 7.36175933796388525523416674195, 9.409944209359452787898729581202, 9.844937907189423851166820201554, 10.63681155504882253053255758163, 12.48548962321777454956813615292, 13.78564306221337635342501044593, 15.31493050615643748662633520979
