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
−13.20655826083354502173669319611, −12.25663443005255035410608295149, −11.64393504455142525260526903332, −10.57920349812506708284830443663, −9.493043455340591183612615818124, −8.261774386261746285142134039193, −6.77046716543358281810016910625, −5.26907339416672429637511452799, −3.29737413779960915996577803214, −1.24436369722393682277219340762,
3.44344665494920354110914738621, 5.38616746346163761481046992650, 6.51355595457610720503687284729, 7.57352000602141638142322471258, 8.710641315208508450370328565584, 10.01309362363580033974786240086, 11.24477655347772771008498266068, 12.08886587580250774595716009420, 13.71293412742061633195454883624, 14.96798869801990113073313013675