| L(s) = 1 | + (0.857 − 2.63i)2-s + (−0.978 − 0.207i)3-s + (−4.61 − 3.35i)4-s + (0.980 + 1.69i)5-s + (−1.38 + 2.40i)6-s + (2.54 − 1.13i)7-s + (−8.31 + 6.03i)8-s + (0.913 + 0.406i)9-s + (5.32 − 1.13i)10-s + (0.173 − 1.65i)11-s + (3.81 + 4.23i)12-s + (1.03 − 1.15i)13-s + (−0.806 − 7.67i)14-s + (−0.605 − 1.86i)15-s + (5.28 + 16.2i)16-s + (0.388 + 3.69i)17-s + ⋯ |
| L(s) = 1 | + (0.606 − 1.86i)2-s + (−0.564 − 0.120i)3-s + (−2.30 − 1.67i)4-s + (0.438 + 0.759i)5-s + (−0.566 + 0.981i)6-s + (0.960 − 0.427i)7-s + (−2.93 + 2.13i)8-s + (0.304 + 0.135i)9-s + (1.68 − 0.357i)10-s + (0.0523 − 0.498i)11-s + (1.10 + 1.22i)12-s + (0.287 − 0.319i)13-s + (−0.215 − 2.05i)14-s + (−0.156 − 0.481i)15-s + (1.32 + 4.06i)16-s + (0.0942 + 0.897i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.809 + 0.586i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.809 + 0.586i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.335302 - 1.03461i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.335302 - 1.03461i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.978 + 0.207i)T \) |
| 31 | \( 1 + (-5.08 + 2.26i)T \) |
| good | 2 | \( 1 + (-0.857 + 2.63i)T + (-1.61 - 1.17i)T^{2} \) |
| 5 | \( 1 + (-0.980 - 1.69i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (-2.54 + 1.13i)T + (4.68 - 5.20i)T^{2} \) |
| 11 | \( 1 + (-0.173 + 1.65i)T + (-10.7 - 2.28i)T^{2} \) |
| 13 | \( 1 + (-1.03 + 1.15i)T + (-1.35 - 12.9i)T^{2} \) |
| 17 | \( 1 + (-0.388 - 3.69i)T + (-16.6 + 3.53i)T^{2} \) |
| 19 | \( 1 + (-0.159 - 0.177i)T + (-1.98 + 18.8i)T^{2} \) |
| 23 | \( 1 + (3.58 - 2.60i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (2.31 - 7.12i)T + (-23.4 - 17.0i)T^{2} \) |
| 37 | \( 1 + (4.24 - 7.35i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (1.07 - 0.228i)T + (37.4 - 16.6i)T^{2} \) |
| 43 | \( 1 + (1.76 + 1.96i)T + (-4.49 + 42.7i)T^{2} \) |
| 47 | \( 1 + (2.22 + 6.84i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (6.35 + 2.82i)T + (35.4 + 39.3i)T^{2} \) |
| 59 | \( 1 + (0.150 + 0.0319i)T + (53.8 + 23.9i)T^{2} \) |
| 61 | \( 1 + 3.56T + 61T^{2} \) |
| 67 | \( 1 + (6.27 + 10.8i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-6.27 - 2.79i)T + (47.5 + 52.7i)T^{2} \) |
| 73 | \( 1 + (-0.660 + 6.28i)T + (-71.4 - 15.1i)T^{2} \) |
| 79 | \( 1 + (-0.510 - 4.85i)T + (-77.2 + 16.4i)T^{2} \) |
| 83 | \( 1 + (3.97 - 0.844i)T + (75.8 - 33.7i)T^{2} \) |
| 89 | \( 1 + (-0.729 - 0.530i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-0.119 - 0.0868i)T + (29.9 + 92.2i)T^{2} \) |
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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
−13.50671598270468952390047882482, −12.34207870382039962248505200774, −11.36255833669027899878703830631, −10.69515451475359541893742948676, −10.00432209231768317036422890637, −8.379902681921591295467639314486, −6.12700594736362488548033300553, −4.90027470912776573697057307842, −3.44622374955185270452636374545, −1.64901289622423240858380564642,
4.42622247593390208970488339477, 5.19243623444242939849953490544, 6.20563092823523504015306283167, 7.53218922793372797359375089660, 8.622035732280866241741772946430, 9.597608527905203382325677912848, 11.75502777759808090251450045817, 12.67134899192001437178482794649, 13.73161741416692368015499623629, 14.55489249009322941252945454898
