| L(s) = 1 | + (−0.841 + 0.540i)2-s + (−0.142 + 0.989i)3-s + (0.415 − 0.909i)4-s + (−4.24 + 1.24i)5-s + (−0.415 − 0.909i)6-s + (−2.17 − 2.50i)7-s + (0.142 + 0.989i)8-s + (−0.959 − 0.281i)9-s + (2.89 − 3.34i)10-s + (1.70 + 1.09i)11-s + (0.841 + 0.540i)12-s + (−1.98 + 2.28i)13-s + (3.18 + 0.934i)14-s + (−0.629 − 4.38i)15-s + (−0.654 − 0.755i)16-s + (−1.24 − 2.73i)17-s + ⋯ |
| L(s) = 1 | + (−0.594 + 0.382i)2-s + (−0.0821 + 0.571i)3-s + (0.207 − 0.454i)4-s + (−1.89 + 0.557i)5-s + (−0.169 − 0.371i)6-s + (−0.820 − 0.947i)7-s + (0.0503 + 0.349i)8-s + (−0.319 − 0.0939i)9-s + (0.916 − 1.05i)10-s + (0.512 + 0.329i)11-s + (0.242 + 0.156i)12-s + (−0.549 + 0.634i)13-s + (0.850 + 0.249i)14-s + (−0.162 − 1.13i)15-s + (−0.163 − 0.188i)16-s + (−0.303 − 0.663i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.975 + 0.221i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.975 + 0.221i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0206107 - 0.183450i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0206107 - 0.183450i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.841 - 0.540i)T \) |
| 3 | \( 1 + (0.142 - 0.989i)T \) |
| 23 | \( 1 + (4.08 - 2.50i)T \) |
| good | 5 | \( 1 + (4.24 - 1.24i)T + (4.20 - 2.70i)T^{2} \) |
| 7 | \( 1 + (2.17 + 2.50i)T + (-0.996 + 6.92i)T^{2} \) |
| 11 | \( 1 + (-1.70 - 1.09i)T + (4.56 + 10.0i)T^{2} \) |
| 13 | \( 1 + (1.98 - 2.28i)T + (-1.85 - 12.8i)T^{2} \) |
| 17 | \( 1 + (1.24 + 2.73i)T + (-11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (2.97 - 6.50i)T + (-12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (-0.633 - 1.38i)T + (-18.9 + 21.9i)T^{2} \) |
| 31 | \( 1 + (-0.822 - 5.72i)T + (-29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (-1.77 - 0.520i)T + (31.1 + 20.0i)T^{2} \) |
| 41 | \( 1 + (-1.02 + 0.299i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (-0.268 + 1.87i)T + (-41.2 - 12.1i)T^{2} \) |
| 47 | \( 1 + 0.748T + 47T^{2} \) |
| 53 | \( 1 + (8.36 + 9.65i)T + (-7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (1.17 - 1.35i)T + (-8.39 - 58.3i)T^{2} \) |
| 61 | \( 1 + (0.556 + 3.86i)T + (-58.5 + 17.1i)T^{2} \) |
| 67 | \( 1 + (3.11 - 2.00i)T + (27.8 - 60.9i)T^{2} \) |
| 71 | \( 1 + (3.36 - 2.16i)T + (29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (2.24 - 4.91i)T + (-47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (5.14 - 5.94i)T + (-11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (-3.87 - 1.13i)T + (69.8 + 44.8i)T^{2} \) |
| 89 | \( 1 + (-0.796 + 5.54i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (12.6 - 3.72i)T + (81.6 - 52.4i)T^{2} \) |
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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
−14.18971534139656807829870684749, −12.36783866729182558499924299840, −11.55227564836671425704228487911, −10.55453383283938563829204998309, −9.716579463076789765267603258389, −8.354993670288463150456079317386, −7.33077999867856119917262433037, −6.59380483939076328050724322401, −4.42291621007127120487084140171, −3.52852355350354608056590196177,
0.22141581082496152748835182267, 2.87748543073863833351537425597, 4.35550002005864244028963604480, 6.29738499178763940358461595920, 7.57001858081612253731377142686, 8.439316581296566524748930220747, 9.213964495524543426966499868871, 10.88826545521193381375979230944, 11.77427481241311381480878970427, 12.42438255609429546976117413770
