| L(s) = 1 | + (3.99 + 0.0841i)2-s + (14.9 + 6.81i)3-s + (15.9 + 0.672i)4-s + (−23.3 + 26.9i)5-s + (59.0 + 28.5i)6-s + (−36.5 − 56.8i)7-s + (63.8 + 4.03i)8-s + (123. + 142. i)9-s + (−95.6 + 105. i)10-s + (118. + 16.9i)11-s + (233. + 118. i)12-s + (−111. − 71.6i)13-s + (−141. − 230. i)14-s + (−532. + 243. i)15-s + (255. + 21.5i)16-s + (−103. − 30.3i)17-s + ⋯ |
| L(s) = 1 | + (0.999 + 0.0210i)2-s + (1.65 + 0.757i)3-s + (0.999 + 0.0420i)4-s + (−0.934 + 1.07i)5-s + (1.64 + 0.791i)6-s + (−0.746 − 1.16i)7-s + (0.998 + 0.0630i)8-s + (1.52 + 1.75i)9-s + (−0.956 + 1.05i)10-s + (0.976 + 0.140i)11-s + (1.62 + 0.826i)12-s + (−0.659 − 0.423i)13-s + (−0.721 − 1.17i)14-s + (−2.36 + 1.08i)15-s + (0.996 + 0.0840i)16-s + (−0.357 − 0.105i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.619 - 0.784i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.619 - 0.784i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(3.92071 + 1.89894i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.92071 + 1.89894i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-3.99 - 0.0841i)T \) |
| 23 | \( 1 + (-477. - 228. i)T \) |
| good | 3 | \( 1 + (-14.9 - 6.81i)T + (53.0 + 61.2i)T^{2} \) |
| 5 | \( 1 + (23.3 - 26.9i)T + (-88.9 - 618. i)T^{2} \) |
| 7 | \( 1 + (36.5 + 56.8i)T + (-997. + 2.18e3i)T^{2} \) |
| 11 | \( 1 + (-118. - 16.9i)T + (1.40e4 + 4.12e3i)T^{2} \) |
| 13 | \( 1 + (111. + 71.6i)T + (1.18e4 + 2.59e4i)T^{2} \) |
| 17 | \( 1 + (103. + 30.3i)T + (7.02e4 + 4.51e4i)T^{2} \) |
| 19 | \( 1 + (165. + 562. i)T + (-1.09e5 + 7.04e4i)T^{2} \) |
| 29 | \( 1 + (389. + 114. i)T + (5.95e5 + 3.82e5i)T^{2} \) |
| 31 | \( 1 + (695. - 317. i)T + (6.04e5 - 6.97e5i)T^{2} \) |
| 37 | \( 1 + (428. + 494. i)T + (-2.66e5 + 1.85e6i)T^{2} \) |
| 41 | \( 1 + (-1.62e3 + 1.87e3i)T + (-4.02e5 - 2.79e6i)T^{2} \) |
| 43 | \( 1 + (-781. - 357. i)T + (2.23e6 + 2.58e6i)T^{2} \) |
| 47 | \( 1 + 1.21e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + (1.63e3 - 1.04e3i)T + (3.27e6 - 7.17e6i)T^{2} \) |
| 59 | \( 1 + (2.52e3 - 3.92e3i)T + (-5.03e6 - 1.10e7i)T^{2} \) |
| 61 | \( 1 + (-272. - 596. i)T + (-9.06e6 + 1.04e7i)T^{2} \) |
| 67 | \( 1 + (2.62e3 - 377. i)T + (1.93e7 - 5.67e6i)T^{2} \) |
| 71 | \( 1 + (1.09e3 - 156. i)T + (2.43e7 - 7.15e6i)T^{2} \) |
| 73 | \( 1 + (3.63e3 - 1.06e3i)T + (2.38e7 - 1.53e7i)T^{2} \) |
| 79 | \( 1 + (2.82e3 - 4.40e3i)T + (-1.61e7 - 3.54e7i)T^{2} \) |
| 83 | \( 1 + (6.89e3 - 5.97e3i)T + (6.75e6 - 4.69e7i)T^{2} \) |
| 89 | \( 1 + (1.24e3 - 2.73e3i)T + (-4.10e7 - 4.74e7i)T^{2} \) |
| 97 | \( 1 + (-5.76e3 + 6.64e3i)T + (-1.25e7 - 8.76e7i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.77817080668384863966494459043, −12.86366840648491632025453535998, −11.16398903107534607734801191763, −10.40213078578732313732811912782, −9.131552612776793016400618192074, −7.34156160935047462743898452457, −7.09547239597259172699519587955, −4.37515374787848774522813832249, −3.61112833067275136803536184085, −2.73107615394889927983397213176,
1.73839682941053234701979267529, 3.18515110559274156739503606626, 4.30394167083605863565903505021, 6.26740190269724170658492853446, 7.55717521052824079590012347500, 8.613749417738660442924442567911, 9.419535136674570563846701386812, 11.76811189328433772675819195127, 12.61561723537260864851595376619, 12.86695085384643614647504224625
