| L(s) = 1 | + (−0.892 + 1.09i)2-s + (1.15 + 2.52i)3-s + (−0.405 − 1.95i)4-s + (1.70 + 1.96i)5-s + (−3.79 − 0.989i)6-s + (−0.650 − 0.418i)7-s + (2.51 + 1.30i)8-s + (−3.07 + 3.54i)9-s + (−3.67 + 0.113i)10-s + (3.87 − 0.557i)11-s + (4.47 − 3.28i)12-s + (−2.44 − 3.80i)13-s + (1.03 − 0.340i)14-s + (−2.99 + 6.56i)15-s + (−3.67 + 1.58i)16-s + (−1.65 − 5.63i)17-s + ⋯ |
| L(s) = 1 | + (−0.631 + 0.775i)2-s + (0.665 + 1.45i)3-s + (−0.202 − 0.979i)4-s + (0.762 + 0.879i)5-s + (−1.54 − 0.403i)6-s + (−0.245 − 0.158i)7-s + (0.887 + 0.460i)8-s + (−1.02 + 1.18i)9-s + (−1.16 + 0.0357i)10-s + (1.16 − 0.168i)11-s + (1.29 − 0.946i)12-s + (−0.678 − 1.05i)13-s + (0.277 − 0.0909i)14-s + (−0.774 + 1.69i)15-s + (−0.917 + 0.397i)16-s + (−0.401 − 1.36i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.672 - 0.740i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.672 - 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.456621 + 1.03155i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.456621 + 1.03155i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.892 - 1.09i)T \) |
| 23 | \( 1 + (4.46 - 1.74i)T \) |
| good | 3 | \( 1 + (-1.15 - 2.52i)T + (-1.96 + 2.26i)T^{2} \) |
| 5 | \( 1 + (-1.70 - 1.96i)T + (-0.711 + 4.94i)T^{2} \) |
| 7 | \( 1 + (0.650 + 0.418i)T + (2.90 + 6.36i)T^{2} \) |
| 11 | \( 1 + (-3.87 + 0.557i)T + (10.5 - 3.09i)T^{2} \) |
| 13 | \( 1 + (2.44 + 3.80i)T + (-5.40 + 11.8i)T^{2} \) |
| 17 | \( 1 + (1.65 + 5.63i)T + (-14.3 + 9.19i)T^{2} \) |
| 19 | \( 1 + (0.786 - 2.67i)T + (-15.9 - 10.2i)T^{2} \) |
| 29 | \( 1 + (1.67 + 5.69i)T + (-24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (-8.29 - 3.78i)T + (20.3 + 23.4i)T^{2} \) |
| 37 | \( 1 + (0.795 - 0.917i)T + (-5.26 - 36.6i)T^{2} \) |
| 41 | \( 1 + (-3.17 - 3.66i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (-0.106 + 0.0487i)T + (28.1 - 32.4i)T^{2} \) |
| 47 | \( 1 - 3.53iT - 47T^{2} \) |
| 53 | \( 1 + (-8.82 - 5.66i)T + (22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (-8.53 + 5.48i)T + (24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (0.0350 - 0.0768i)T + (-39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (7.69 + 1.10i)T + (64.2 + 18.8i)T^{2} \) |
| 71 | \( 1 + (1.81 + 0.261i)T + (68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (6.54 + 1.92i)T + (61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (11.5 - 7.43i)T + (32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (1.04 + 0.909i)T + (11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (1.28 - 0.586i)T + (58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + (11.7 - 10.1i)T + (13.8 - 96.0i)T^{2} \) |
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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
−13.66728350142370158975259218604, −11.58121478513949849230919247704, −10.28494845316012272463225978919, −9.967516965618387463723418517503, −9.224899030359721527018269400670, −8.080556430001020439049114583481, −6.76551125662206686517476909885, −5.65873449593635456901726085144, −4.31402426834689491021792079473, −2.75312422398084649400622548531,
1.43294423631127870235359638849, 2.30506286401791032176483212488, 4.20382159968292882234394334952, 6.31202265958183569691971767848, 7.23045238015233370161594285062, 8.605209938593257486169731572718, 8.958817202801029617762656639533, 10.01049643554367382949193333551, 11.65208516189844606920677069835, 12.35005007945272869432885300438
