| L(s) = 1 | + (0.654 − 0.755i)2-s + (0.841 − 0.540i)3-s + (−0.142 − 0.989i)4-s + (0.415 + 0.909i)5-s + (0.142 − 0.989i)6-s + (1.88 − 0.554i)7-s + (−0.841 − 0.540i)8-s + (0.415 − 0.909i)9-s + (0.959 + 0.281i)10-s + (−1.64 − 1.89i)11-s + (−0.654 − 0.755i)12-s + (2.73 + 0.803i)13-s + (0.817 − 1.78i)14-s + (0.841 + 0.540i)15-s + (−0.959 + 0.281i)16-s + (0.577 − 4.01i)17-s + ⋯ |
| L(s) = 1 | + (0.463 − 0.534i)2-s + (0.485 − 0.312i)3-s + (−0.0711 − 0.494i)4-s + (0.185 + 0.406i)5-s + (0.0580 − 0.404i)6-s + (0.713 − 0.209i)7-s + (−0.297 − 0.191i)8-s + (0.138 − 0.303i)9-s + (0.303 + 0.0890i)10-s + (−0.496 − 0.572i)11-s + (−0.189 − 0.218i)12-s + (0.758 + 0.222i)13-s + (0.218 − 0.478i)14-s + (0.217 + 0.139i)15-s + (−0.239 + 0.0704i)16-s + (0.140 − 0.974i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.257 + 0.966i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.257 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.94073 - 1.49102i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.94073 - 1.49102i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.654 + 0.755i)T \) |
| 3 | \( 1 + (-0.841 + 0.540i)T \) |
| 5 | \( 1 + (-0.415 - 0.909i)T \) |
| 23 | \( 1 + (-4.06 + 2.53i)T \) |
| good | 7 | \( 1 + (-1.88 + 0.554i)T + (5.88 - 3.78i)T^{2} \) |
| 11 | \( 1 + (1.64 + 1.89i)T + (-1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-2.73 - 0.803i)T + (10.9 + 7.02i)T^{2} \) |
| 17 | \( 1 + (-0.577 + 4.01i)T + (-16.3 - 4.78i)T^{2} \) |
| 19 | \( 1 + (-0.304 - 2.11i)T + (-18.2 + 5.35i)T^{2} \) |
| 29 | \( 1 + (0.205 - 1.43i)T + (-27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (1.35 + 0.868i)T + (12.8 + 28.1i)T^{2} \) |
| 37 | \( 1 + (1.24 - 2.72i)T + (-24.2 - 27.9i)T^{2} \) |
| 41 | \( 1 + (-1.12 - 2.45i)T + (-26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (-5.39 + 3.46i)T + (17.8 - 39.1i)T^{2} \) |
| 47 | \( 1 + 4.91T + 47T^{2} \) |
| 53 | \( 1 + (2.96 - 0.871i)T + (44.5 - 28.6i)T^{2} \) |
| 59 | \( 1 + (9.29 + 2.73i)T + (49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (-9.75 - 6.26i)T + (25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (9.41 - 10.8i)T + (-9.53 - 66.3i)T^{2} \) |
| 71 | \( 1 + (-5.88 + 6.79i)T + (-10.1 - 70.2i)T^{2} \) |
| 73 | \( 1 + (-0.0635 - 0.442i)T + (-70.0 + 20.5i)T^{2} \) |
| 79 | \( 1 + (6.00 + 1.76i)T + (66.4 + 42.7i)T^{2} \) |
| 83 | \( 1 + (4.66 - 10.2i)T + (-54.3 - 62.7i)T^{2} \) |
| 89 | \( 1 + (3.72 - 2.39i)T + (36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + (-4.10 - 8.98i)T + (-63.5 + 73.3i)T^{2} \) |
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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
−10.52713020662825951487008856988, −9.476204506148550267858172609662, −8.594639339327500770645298196944, −7.70562114685887308900111192768, −6.72966568125204480269196174134, −5.69537759873226292882801112563, −4.67789024030097800379928874715, −3.48101259362251422816250050913, −2.57393247018989820754259413331, −1.24589263447782183580868591185,
1.78549603463505712706826650956, 3.20137956328753482969692114537, 4.35102883649460958423872274659, 5.14193009237275887727750630592, 6.01597898005373954710573766216, 7.26338377359039671312257315855, 8.086601892117494254303156214123, 8.737387194956115947571615993195, 9.606014391535896589017417318075, 10.69680473404320479642930542612
