| L(s) = 1 | + (0.870 − 1.19i)2-s + (0.0776 − 1.73i)3-s + (−0.0590 − 0.181i)4-s + (−1.02 + 0.590i)5-s + (−2.00 − 1.59i)6-s + (0.114 + 0.0243i)7-s + (2.54 + 0.827i)8-s + (−2.98 − 0.268i)9-s + (−0.182 + 1.73i)10-s + (−1.65 + 1.83i)11-s + (−0.318 + 0.0880i)12-s + (−0.591 + 1.32i)13-s + (0.128 − 0.115i)14-s + (0.942 + 1.81i)15-s + (3.51 − 2.55i)16-s + (−0.129 − 0.144i)17-s + ⋯ |
| L(s) = 1 | + (0.615 − 0.846i)2-s + (0.0448 − 0.998i)3-s + (−0.0295 − 0.0908i)4-s + (−0.457 + 0.264i)5-s + (−0.818 − 0.652i)6-s + (0.0432 + 0.00920i)7-s + (0.900 + 0.292i)8-s + (−0.995 − 0.0895i)9-s + (−0.0577 + 0.549i)10-s + (−0.498 + 0.553i)11-s + (−0.0920 + 0.0254i)12-s + (−0.163 + 0.368i)13-s + (0.0344 − 0.0310i)14-s + (0.243 + 0.468i)15-s + (0.878 − 0.638i)16-s + (−0.0314 − 0.0349i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.163 + 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.163 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.975189 - 0.826920i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.975189 - 0.826920i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.0776 + 1.73i)T \) |
| 31 | \( 1 + (1.70 + 5.30i)T \) |
| good | 2 | \( 1 + (-0.870 + 1.19i)T + (-0.618 - 1.90i)T^{2} \) |
| 5 | \( 1 + (1.02 - 0.590i)T + (2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (-0.114 - 0.0243i)T + (6.39 + 2.84i)T^{2} \) |
| 11 | \( 1 + (1.65 - 1.83i)T + (-1.14 - 10.9i)T^{2} \) |
| 13 | \( 1 + (0.591 - 1.32i)T + (-8.69 - 9.66i)T^{2} \) |
| 17 | \( 1 + (0.129 + 0.144i)T + (-1.77 + 16.9i)T^{2} \) |
| 19 | \( 1 + (-5.86 + 2.61i)T + (12.7 - 14.1i)T^{2} \) |
| 23 | \( 1 + (1.06 - 3.27i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (7.42 + 5.39i)T + (8.96 + 27.5i)T^{2} \) |
| 37 | \( 1 + (7.34 + 4.24i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-3.19 - 0.336i)T + (40.1 + 8.52i)T^{2} \) |
| 43 | \( 1 + (-2.97 - 6.68i)T + (-28.7 + 31.9i)T^{2} \) |
| 47 | \( 1 + (2.01 + 2.77i)T + (-14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (-6.25 + 1.33i)T + (48.4 - 21.5i)T^{2} \) |
| 59 | \( 1 + (-7.45 + 0.783i)T + (57.7 - 12.2i)T^{2} \) |
| 61 | \( 1 + 11.2iT - 61T^{2} \) |
| 67 | \( 1 + (3.96 + 6.86i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-3.31 - 15.6i)T + (-64.8 + 28.8i)T^{2} \) |
| 73 | \( 1 + (-6.58 - 5.93i)T + (7.63 + 72.6i)T^{2} \) |
| 79 | \( 1 + (-1.39 + 1.25i)T + (8.25 - 78.5i)T^{2} \) |
| 83 | \( 1 + (0.267 - 2.54i)T + (-81.1 - 17.2i)T^{2} \) |
| 89 | \( 1 + (3.58 + 11.0i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (-2.44 - 7.53i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−13.44524181698741201281820109081, −12.79008755105439809653324603987, −11.62339584629647582933318627558, −11.29588933555634973251234148974, −9.590214270762611413491484733352, −7.81962194867781361553495874241, −7.22827286537235038787170406347, −5.35176927060955710214338302594, −3.60630162777576220001044961015, −2.14233100406250546326457044883,
3.57760782118376600023137945986, 4.95521771164327563986657963111, 5.79059961273062025169511109805, 7.45469118001524494954621188542, 8.601167168484512483463917830556, 10.05560576058154738085515816152, 10.92058932394706752900994801804, 12.26038448608685486519157137904, 13.67349292610545558034371271656, 14.45409485486782836895915360602
