L(s) = 1 | + (1.29 + 2.24i)2-s + (−1.72 + 0.120i)3-s + (−2.36 + 4.09i)4-s + (−0.137 − 0.0793i)5-s + (−2.51 − 3.72i)6-s + (2.91 − 1.68i)7-s − 7.08·8-s + (2.97 − 0.416i)9-s − 0.411i·10-s + (−1.95 + 2.68i)11-s + (3.59 − 7.36i)12-s + (0.897 + 0.518i)13-s + (7.57 + 4.37i)14-s + (0.246 + 0.120i)15-s + (−4.45 − 7.72i)16-s + 5.02·17-s + ⋯ |
L(s) = 1 | + (0.917 + 1.58i)2-s + (−0.997 + 0.0695i)3-s + (−1.18 + 2.04i)4-s + (−0.0614 − 0.0354i)5-s + (−1.02 − 1.52i)6-s + (1.10 − 0.636i)7-s − 2.50·8-s + (0.990 − 0.138i)9-s − 0.130i·10-s + (−0.588 + 0.808i)11-s + (1.03 − 2.12i)12-s + (0.249 + 0.143i)13-s + (2.02 + 1.16i)14-s + (0.0637 + 0.0311i)15-s + (−1.11 − 1.93i)16-s + 1.21·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.668 - 0.743i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.668 - 0.743i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.477922 + 1.07185i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.477922 + 1.07185i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.72 - 0.120i)T \) |
| 11 | \( 1 + (1.95 - 2.68i)T \) |
good | 2 | \( 1 + (-1.29 - 2.24i)T + (-1 + 1.73i)T^{2} \) |
| 5 | \( 1 + (0.137 + 0.0793i)T + (2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (-2.91 + 1.68i)T + (3.5 - 6.06i)T^{2} \) |
| 13 | \( 1 + (-0.897 - 0.518i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 5.02T + 17T^{2} \) |
| 19 | \( 1 + 5.11iT - 19T^{2} \) |
| 23 | \( 1 + (4.02 + 2.32i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (0.673 + 1.16i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-1.62 + 2.81i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 3.04T + 37T^{2} \) |
| 41 | \( 1 + (0.940 - 1.62i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (8.24 - 4.76i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (0.996 - 0.575i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 9.36iT - 53T^{2} \) |
| 59 | \( 1 + (4.69 + 2.71i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-9.61 + 5.55i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (2.67 - 4.64i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 0.558iT - 71T^{2} \) |
| 73 | \( 1 - 6.24iT - 73T^{2} \) |
| 79 | \( 1 + (-3.72 + 2.14i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (4.61 + 7.99i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 1.98iT - 89T^{2} \) |
| 97 | \( 1 + (-0.917 - 1.58i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−14.43489841901467949884094920712, −13.46171895216187348947564522207, −12.46022080347271610453495439378, −11.44719900613034829498678711721, −10.01229191452708148312311244347, −8.091685365324316533218571387554, −7.36045106199882077843366934936, −6.18412312410294999494213989015, −4.98513744527290529574655120791, −4.28918339287239749392624207870,
1.62087874258662038694229477624, 3.63475870494849995972411398653, 5.21733399373298807895685558511, 5.72979999742783073331593532163, 8.090886099536299875947546679846, 9.881260514804053982551502044486, 10.73135089418730866957309234702, 11.65058193899818675990721769526, 12.09334466751631003468149545522, 13.20414603131575650523768542926