| L(s) = 1 | + (0.253 − 5.65i)2-s + (−2.08 − 15.4i)3-s + (−31.8 − 2.86i)4-s + (84.3 − 48.6i)5-s + (−87.8 + 7.87i)6-s + (−87.9 − 50.7i)7-s + (−24.2 + 179. i)8-s + (−234. + 64.5i)9-s + (−253. − 488. i)10-s + (−73.2 + 126. i)11-s + (22.2 + 498. i)12-s + (402. + 697. i)13-s + (−309. + 484. i)14-s + (−928. − 1.20e3i)15-s + (1.00e3 + 182. i)16-s − 1.23e3i·17-s + ⋯ |
| L(s) = 1 | + (0.0448 − 0.998i)2-s + (−0.133 − 0.990i)3-s + (−0.995 − 0.0895i)4-s + (1.50 − 0.870i)5-s + (−0.995 + 0.0893i)6-s + (−0.678 − 0.391i)7-s + (−0.134 + 0.990i)8-s + (−0.964 + 0.265i)9-s + (−0.802 − 1.54i)10-s + (−0.182 + 0.316i)11-s + (0.0446 + 0.999i)12-s + (0.661 + 1.14i)13-s + (−0.421 + 0.660i)14-s + (−1.06 − 1.37i)15-s + (0.983 + 0.178i)16-s − 1.03i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.00442i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.00442i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.00325425 - 1.47071i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00325425 - 1.47071i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.253 + 5.65i)T \) |
| 3 | \( 1 + (2.08 + 15.4i)T \) |
| good | 5 | \( 1 + (-84.3 + 48.6i)T + (1.56e3 - 2.70e3i)T^{2} \) |
| 7 | \( 1 + (87.9 + 50.7i)T + (8.40e3 + 1.45e4i)T^{2} \) |
| 11 | \( 1 + (73.2 - 126. i)T + (-8.05e4 - 1.39e5i)T^{2} \) |
| 13 | \( 1 + (-402. - 697. i)T + (-1.85e5 + 3.21e5i)T^{2} \) |
| 17 | \( 1 + 1.23e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 2.61e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 + (-84.5 - 146. i)T + (-3.21e6 + 5.57e6i)T^{2} \) |
| 29 | \( 1 + (-1.95e3 - 1.12e3i)T + (1.02e7 + 1.77e7i)T^{2} \) |
| 31 | \( 1 + (-1.78e3 + 1.03e3i)T + (1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 - 445.T + 6.93e7T^{2} \) |
| 41 | \( 1 + (1.72e3 - 993. i)T + (5.79e7 - 1.00e8i)T^{2} \) |
| 43 | \( 1 + (-3.46e3 - 2.00e3i)T + (7.35e7 + 1.27e8i)T^{2} \) |
| 47 | \( 1 + (3.53e3 - 6.13e3i)T + (-1.14e8 - 1.98e8i)T^{2} \) |
| 53 | \( 1 + 6.23e3iT - 4.18e8T^{2} \) |
| 59 | \( 1 + (-1.67e4 - 2.89e4i)T + (-3.57e8 + 6.19e8i)T^{2} \) |
| 61 | \( 1 + (-1.47e4 + 2.54e4i)T + (-4.22e8 - 7.31e8i)T^{2} \) |
| 67 | \( 1 + (-2.33e4 + 1.34e4i)T + (6.75e8 - 1.16e9i)T^{2} \) |
| 71 | \( 1 - 3.63e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.82e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + (-5.45e4 - 3.14e4i)T + (1.53e9 + 2.66e9i)T^{2} \) |
| 83 | \( 1 + (3.99e4 - 6.92e4i)T + (-1.96e9 - 3.41e9i)T^{2} \) |
| 89 | \( 1 - 9.66e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 + (8.31e3 - 1.44e4i)T + (-4.29e9 - 7.43e9i)T^{2} \) |
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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
−13.91698803834709081369670402044, −13.48484349805511634401198517257, −12.64476190826840007388539018006, −11.31363762885744398960569072177, −9.697941368874869018479656528184, −8.846018238637921171755533173385, −6.59865378391320494867172433196, −5.03173943400787595111675048053, −2.38151289511955541999131415622, −0.919698606723005918788729132289,
3.35423089345665393788318866146, 5.69155234221437689441890045432, 6.17613195403973806487377728619, 8.460649071955772753869157284114, 9.872162616360167509084042032139, 10.44735386386255687357645294664, 12.85467647481158788502007812634, 14.04155564277456395052749978232, 14.92874256893963954590231190239, 15.95164915714940971490263413000
