| L(s) = 1 | + (−49.0 + 8.64i)2-s + (84.6 + 100. i)3-s + (1.36e3 − 498. i)4-s + (−1.31e3 − 477. i)5-s + (−5.02e3 − 4.21e3i)6-s + (−1.26e4 + 2.18e4i)7-s + (−1.86e4 + 1.07e4i)8-s + (7.24e3 − 4.10e4i)9-s + (6.85e4 + 1.20e4i)10-s + (−8.63e4 − 1.49e5i)11-s + (1.66e5 + 9.59e4i)12-s + (−3.33e4 + 3.96e4i)13-s + (4.30e5 − 1.18e6i)14-s + (−6.29e4 − 1.72e5i)15-s + (−3.19e5 + 2.68e5i)16-s + (−3.28e4 − 1.86e5i)17-s + ⋯ |
| L(s) = 1 | + (−1.53 + 0.270i)2-s + (0.348 + 0.414i)3-s + (1.33 − 0.486i)4-s + (−0.420 − 0.152i)5-s + (−0.645 − 0.542i)6-s + (−0.751 + 1.30i)7-s + (−0.570 + 0.329i)8-s + (0.122 − 0.695i)9-s + (0.685 + 0.120i)10-s + (−0.536 − 0.928i)11-s + (0.667 + 0.385i)12-s + (−0.0897 + 0.106i)13-s + (0.800 − 2.19i)14-s + (−0.0828 − 0.227i)15-s + (−0.305 + 0.255i)16-s + (−0.0231 − 0.131i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.238 - 0.971i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.238 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(0.406353 + 0.318722i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.406353 + 0.318722i\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (1.31e3 + 477. i)T \) |
| 19 | \( 1 + (1.75e6 - 1.74e6i)T \) |
| good | 2 | \( 1 + (49.0 - 8.64i)T + (962. - 350. i)T^{2} \) |
| 3 | \( 1 + (-84.6 - 100. i)T + (-1.02e4 + 5.81e4i)T^{2} \) |
| 7 | \( 1 + (1.26e4 - 2.18e4i)T + (-1.41e8 - 2.44e8i)T^{2} \) |
| 11 | \( 1 + (8.63e4 + 1.49e5i)T + (-1.29e10 + 2.24e10i)T^{2} \) |
| 13 | \( 1 + (3.33e4 - 3.96e4i)T + (-2.39e10 - 1.35e11i)T^{2} \) |
| 17 | \( 1 + (3.28e4 + 1.86e5i)T + (-1.89e12 + 6.89e11i)T^{2} \) |
| 23 | \( 1 + (-4.21e6 + 1.53e6i)T + (3.17e13 - 2.66e13i)T^{2} \) |
| 29 | \( 1 + (2.64e7 + 4.65e6i)T + (3.95e14 + 1.43e14i)T^{2} \) |
| 31 | \( 1 + (-1.09e7 - 6.32e6i)T + (4.09e14 + 7.09e14i)T^{2} \) |
| 37 | \( 1 + 6.98e7iT - 4.80e15T^{2} \) |
| 41 | \( 1 + (6.39e7 + 7.62e7i)T + (-2.33e15 + 1.32e16i)T^{2} \) |
| 43 | \( 1 + (-7.00e6 - 2.54e6i)T + (1.65e16 + 1.38e16i)T^{2} \) |
| 47 | \( 1 + (-4.61e7 + 2.61e8i)T + (-4.94e16 - 1.79e16i)T^{2} \) |
| 53 | \( 1 + (-4.97e7 - 1.36e8i)T + (-1.33e17 + 1.12e17i)T^{2} \) |
| 59 | \( 1 + (-4.90e8 + 8.64e7i)T + (4.80e17 - 1.74e17i)T^{2} \) |
| 61 | \( 1 + (-9.32e8 + 3.39e8i)T + (5.46e17 - 4.58e17i)T^{2} \) |
| 67 | \( 1 + (2.38e8 + 4.20e7i)T + (1.71e18 + 6.23e17i)T^{2} \) |
| 71 | \( 1 + (-2.85e7 + 7.84e7i)T + (-2.49e18 - 2.09e18i)T^{2} \) |
| 73 | \( 1 + (1.29e9 - 1.08e9i)T + (7.46e17 - 4.23e18i)T^{2} \) |
| 79 | \( 1 + (-1.28e9 - 1.53e9i)T + (-1.64e18 + 9.32e18i)T^{2} \) |
| 83 | \( 1 + (1.44e9 - 2.49e9i)T + (-7.75e18 - 1.34e19i)T^{2} \) |
| 89 | \( 1 + (2.35e9 - 2.81e9i)T + (-5.41e18 - 3.07e19i)T^{2} \) |
| 97 | \( 1 + (2.78e9 - 4.90e8i)T + (6.92e19 - 2.52e19i)T^{2} \) |
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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
−11.95400161659744739177038864579, −10.74543067415328340097527385771, −9.647929704338131853371352545027, −8.891111815905309734469688456448, −8.291589702348722627563958989410, −6.85597368132605978086691916889, −5.69658365957751658032782515342, −3.67518368604507614104734359858, −2.33987720353619001466080433254, −0.59692811607579287282926323970,
0.37456040255814362009042658908, 1.62682964240144249320618642198, 2.86966236034112449273006083276, 4.55087764565942553211973774790, 6.96053010660003790180251273908, 7.40813468460206720678182520282, 8.378974384554256651661214658287, 9.693282967044090525662919309977, 10.44129652577384081549605789965, 11.23830893668363690488248254215
