| L(s) = 1 | + (−2.13 − 2.54i)2-s + (−1.22 + 6.92i)4-s + (0.870 + 4.93i)5-s + (2.46 − 4.27i)7-s + (8.70 − 5.02i)8-s + (10.7 − 12.7i)10-s + (−7.89 − 13.6i)11-s + (−4.52 − 12.4i)13-s + (−16.1 + 2.84i)14-s + (−4.96 − 1.80i)16-s + (18.3 − 15.3i)17-s + (−11.5 − 15.0i)19-s − 35.2·20-s + (−17.9 + 49.2i)22-s + (−4.49 + 25.5i)23-s + ⋯ |
| L(s) = 1 | + (−1.06 − 1.27i)2-s + (−0.305 + 1.73i)4-s + (0.174 + 0.987i)5-s + (0.352 − 0.610i)7-s + (1.08 − 0.628i)8-s + (1.07 − 1.27i)10-s + (−0.717 − 1.24i)11-s + (−0.348 − 0.957i)13-s + (−1.15 + 0.203i)14-s + (−0.310 − 0.112i)16-s + (1.07 − 0.904i)17-s + (−0.608 − 0.793i)19-s − 1.76·20-s + (−0.815 + 2.24i)22-s + (−0.195 + 1.10i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.887 + 0.460i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.887 + 0.460i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.157995 - 0.647569i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.157995 - 0.647569i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 19 | \( 1 + (11.5 + 15.0i)T \) |
| good | 2 | \( 1 + (2.13 + 2.54i)T + (-0.694 + 3.93i)T^{2} \) |
| 5 | \( 1 + (-0.870 - 4.93i)T + (-23.4 + 8.55i)T^{2} \) |
| 7 | \( 1 + (-2.46 + 4.27i)T + (-24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (7.89 + 13.6i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (4.52 + 12.4i)T + (-129. + 108. i)T^{2} \) |
| 17 | \( 1 + (-18.3 + 15.3i)T + (50.1 - 284. i)T^{2} \) |
| 23 | \( 1 + (4.49 - 25.5i)T + (-497. - 180. i)T^{2} \) |
| 29 | \( 1 + (-9.60 + 11.4i)T + (-146. - 828. i)T^{2} \) |
| 31 | \( 1 + (32.4 + 18.7i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + 22.7iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (-20.3 + 55.8i)T + (-1.28e3 - 1.08e3i)T^{2} \) |
| 43 | \( 1 + (-9.77 - 55.4i)T + (-1.73e3 + 632. i)T^{2} \) |
| 47 | \( 1 + (58.7 + 49.2i)T + (383. + 2.17e3i)T^{2} \) |
| 53 | \( 1 + (-28.6 - 5.05i)T + (2.63e3 + 960. i)T^{2} \) |
| 59 | \( 1 + (-21.0 - 25.0i)T + (-604. + 3.42e3i)T^{2} \) |
| 61 | \( 1 + (-3.90 + 22.1i)T + (-3.49e3 - 1.27e3i)T^{2} \) |
| 67 | \( 1 + (-6.43 + 7.67i)T + (-779. - 4.42e3i)T^{2} \) |
| 71 | \( 1 + (7.22 - 1.27i)T + (4.73e3 - 1.72e3i)T^{2} \) |
| 73 | \( 1 + (-24.7 - 9.02i)T + (4.08e3 + 3.42e3i)T^{2} \) |
| 79 | \( 1 + (-31.1 + 85.7i)T + (-4.78e3 - 4.01e3i)T^{2} \) |
| 83 | \( 1 + (68.8 - 119. i)T + (-3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + (8.75 + 24.0i)T + (-6.06e3 + 5.09e3i)T^{2} \) |
| 97 | \( 1 + (-78.1 - 93.0i)T + (-1.63e3 + 9.26e3i)T^{2} \) |
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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
−11.60478667508815851702461706824, −10.88817042290025413158205986734, −10.36885388432589913903933997204, −9.387143604846960208088011370937, −8.096703925663939575240613615730, −7.36027785652795646322536568293, −5.57900035340496868296469546004, −3.47048719668576038628011849436, −2.57099998104238593592653287323, −0.58743369975336484299535905677,
1.68904121127376105414579407844, 4.68086025229813120228182652756, 5.63274821271059504271426711285, 6.84507620729602072456739802851, 8.013474352537609580471227950801, 8.642393391904273445440559125212, 9.614976157436959703404388459633, 10.41741590722385411771521214539, 12.20909098326541267375733262321, 12.76077964305293394087905229321
