| L(s) = 1 | + (−0.483 + 1.48i)2-s + (−0.669 + 0.743i)3-s + (−0.360 − 0.262i)4-s + (−0.686 + 1.18i)5-s + (−0.781 − 1.35i)6-s + (0.0145 − 0.138i)7-s + (−1.96 + 1.42i)8-s + (−0.104 − 0.994i)9-s + (−1.43 − 1.59i)10-s + (0.454 − 0.202i)11-s + (0.436 − 0.0927i)12-s + (2.00 + 0.425i)13-s + (0.199 + 0.0887i)14-s + (−0.424 − 1.30i)15-s + (−1.45 − 4.46i)16-s + (4.82 + 2.14i)17-s + ⋯ |
| L(s) = 1 | + (−0.341 + 1.05i)2-s + (−0.386 + 0.429i)3-s + (−0.180 − 0.131i)4-s + (−0.306 + 0.531i)5-s + (−0.319 − 0.552i)6-s + (0.00551 − 0.0524i)7-s + (−0.695 + 0.505i)8-s + (−0.0348 − 0.331i)9-s + (−0.454 − 0.504i)10-s + (0.136 − 0.0609i)11-s + (0.125 − 0.0267i)12-s + (0.555 + 0.118i)13-s + (0.0532 + 0.0237i)14-s + (−0.109 − 0.337i)15-s + (−0.362 − 1.11i)16-s + (1.17 + 0.520i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.698 - 0.715i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.698 - 0.715i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.292381 + 0.694623i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.292381 + 0.694623i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.669 - 0.743i)T \) |
| 31 | \( 1 + (3.54 + 4.29i)T \) |
| good | 2 | \( 1 + (0.483 - 1.48i)T + (-1.61 - 1.17i)T^{2} \) |
| 5 | \( 1 + (0.686 - 1.18i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (-0.0145 + 0.138i)T + (-6.84 - 1.45i)T^{2} \) |
| 11 | \( 1 + (-0.454 + 0.202i)T + (7.36 - 8.17i)T^{2} \) |
| 13 | \( 1 + (-2.00 - 0.425i)T + (11.8 + 5.28i)T^{2} \) |
| 17 | \( 1 + (-4.82 - 2.14i)T + (11.3 + 12.6i)T^{2} \) |
| 19 | \( 1 + (-2.80 + 0.596i)T + (17.3 - 7.72i)T^{2} \) |
| 23 | \( 1 + (-1.87 + 1.36i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (-1.43 + 4.43i)T + (-23.4 - 17.0i)T^{2} \) |
| 37 | \( 1 + (-2.10 - 3.64i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (7.20 + 7.99i)T + (-4.28 + 40.7i)T^{2} \) |
| 43 | \( 1 + (12.2 - 2.61i)T + (39.2 - 17.4i)T^{2} \) |
| 47 | \( 1 + (3.28 + 10.1i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-1.02 - 9.78i)T + (-51.8 + 11.0i)T^{2} \) |
| 59 | \( 1 + (-2.25 + 2.50i)T + (-6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 + 1.84T + 61T^{2} \) |
| 67 | \( 1 + (2.60 - 4.52i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (0.592 + 5.63i)T + (-69.4 + 14.7i)T^{2} \) |
| 73 | \( 1 + (-10.1 + 4.51i)T + (48.8 - 54.2i)T^{2} \) |
| 79 | \( 1 + (-5.15 - 2.29i)T + (52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (1.42 + 1.58i)T + (-8.67 + 82.5i)T^{2} \) |
| 89 | \( 1 + (-1.96 - 1.42i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (6.84 + 4.97i)T + (29.9 + 92.2i)T^{2} \) |
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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
−14.96798869801990113073313013675, −13.71293412742061633195454883624, −12.08886587580250774595716009420, −11.24477655347772771008498266068, −10.01309362363580033974786240086, −8.710641315208508450370328565584, −7.57352000602141638142322471258, −6.51355595457610720503687284729, −5.38616746346163761481046992650, −3.44344665494920354110914738621,
1.24436369722393682277219340762, 3.29737413779960915996577803214, 5.26907339416672429637511452799, 6.77046716543358281810016910625, 8.261774386261746285142134039193, 9.493043455340591183612615818124, 10.57920349812506708284830443663, 11.64393504455142525260526903332, 12.25663443005255035410608295149, 13.20655826083354502173669319611
