| L(s) = 1 | + (0.0571 + 0.125i)3-s + (1.77 + 2.04i)5-s + (−0.600 − 0.385i)7-s + (1.95 − 2.25i)9-s + (−0.176 − 1.22i)11-s + (−3.37 + 2.16i)13-s + (−0.154 + 0.339i)15-s + (−1.87 + 0.551i)17-s + (−5.89 − 1.73i)19-s + (0.0139 − 0.0970i)21-s + (−1.81 − 4.43i)23-s + (−0.334 + 2.32i)25-s + (0.788 + 0.231i)27-s + (7.07 − 2.07i)29-s + (−1.60 + 3.50i)31-s + ⋯ |
| L(s) = 1 | + (0.0329 + 0.0721i)3-s + (0.794 + 0.916i)5-s + (−0.226 − 0.145i)7-s + (0.650 − 0.750i)9-s + (−0.0532 − 0.370i)11-s + (−0.936 + 0.601i)13-s + (−0.0399 + 0.0875i)15-s + (−0.455 + 0.133i)17-s + (−1.35 − 0.397i)19-s + (0.00304 − 0.0211i)21-s + (−0.379 − 0.925i)23-s + (−0.0669 + 0.465i)25-s + (0.151 + 0.0445i)27-s + (1.31 − 0.385i)29-s + (−0.287 + 0.629i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.960 - 0.279i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.960 - 0.279i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.05464 + 0.150131i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.05464 + 0.150131i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 + (1.81 + 4.43i)T \) |
| good | 3 | \( 1 + (-0.0571 - 0.125i)T + (-1.96 + 2.26i)T^{2} \) |
| 5 | \( 1 + (-1.77 - 2.04i)T + (-0.711 + 4.94i)T^{2} \) |
| 7 | \( 1 + (0.600 + 0.385i)T + (2.90 + 6.36i)T^{2} \) |
| 11 | \( 1 + (0.176 + 1.22i)T + (-10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (3.37 - 2.16i)T + (5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (1.87 - 0.551i)T + (14.3 - 9.19i)T^{2} \) |
| 19 | \( 1 + (5.89 + 1.73i)T + (15.9 + 10.2i)T^{2} \) |
| 29 | \( 1 + (-7.07 + 2.07i)T + (24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (1.60 - 3.50i)T + (-20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 + (1.59 - 1.83i)T + (-5.26 - 36.6i)T^{2} \) |
| 41 | \( 1 + (2.23 + 2.58i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (-4.86 - 10.6i)T + (-28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 - 5.43T + 47T^{2} \) |
| 53 | \( 1 + (-6.61 - 4.24i)T + (22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (-8.60 + 5.53i)T + (24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (4.24 - 9.30i)T + (-39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (-0.870 + 6.05i)T + (-64.2 - 18.8i)T^{2} \) |
| 71 | \( 1 + (2.12 - 14.7i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (14.6 + 4.29i)T + (61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (-8.62 + 5.54i)T + (32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (7.65 - 8.83i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (0.864 + 1.89i)T + (-58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 + (0.587 + 0.678i)T + (-13.8 + 96.0i)T^{2} \) |
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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
−14.24969046075544004724206341702, −13.12590290012436250992102320032, −12.06084272179725520114570932878, −10.62481114485437079258651151627, −9.951694541521346541793754419617, −8.743579642326995747784634685335, −6.93551937923037438871859611355, −6.30324591212410399221291910032, −4.35524182712235341189865885042, −2.53009620560665999772594151532,
2.09251117208412343917962359886, 4.53836426464512776013133190896, 5.66195248193507926683913689134, 7.23934693606977182565454248779, 8.545975262317601823530086809645, 9.712699388648174332472823468707, 10.53880257756127233145203243078, 12.25862290597148064870024709844, 12.94122622852946429941335827943, 13.78008573543634208043727416058
