| L(s) = 1 | + (−0.909 − 1.08i)2-s + (−0.347 + 1.96i)4-s + (−0.387 − 1.06i)5-s + (−0.332 − 1.88i)7-s + (2.44 − 1.41i)8-s + (−0.800 + 1.38i)10-s + (6.05 − 16.6i)11-s + (−9.14 − 7.67i)13-s + (−1.73 + 2.07i)14-s + (−3.75 − 1.36i)16-s + (2.93 + 1.69i)17-s + (−10.2 − 17.7i)19-s + (2.22 − 0.393i)20-s + (−23.5 + 8.56i)22-s + (−0.678 − 0.119i)23-s + ⋯ |
| L(s) = 1 | + (−0.454 − 0.541i)2-s + (−0.0868 + 0.492i)4-s + (−0.0774 − 0.212i)5-s + (−0.0474 − 0.269i)7-s + (0.306 − 0.176i)8-s + (−0.0800 + 0.138i)10-s + (0.550 − 1.51i)11-s + (−0.703 − 0.590i)13-s + (−0.124 + 0.148i)14-s + (−0.234 − 0.0855i)16-s + (0.172 + 0.0998i)17-s + (−0.538 − 0.932i)19-s + (0.111 − 0.0196i)20-s + (−1.06 + 0.389i)22-s + (−0.0294 − 0.00519i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.469 + 0.883i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.469 + 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.487832 - 0.811505i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.487832 - 0.811505i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.909 + 1.08i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (0.387 + 1.06i)T + (-19.1 + 16.0i)T^{2} \) |
| 7 | \( 1 + (0.332 + 1.88i)T + (-46.0 + 16.7i)T^{2} \) |
| 11 | \( 1 + (-6.05 + 16.6i)T + (-92.6 - 77.7i)T^{2} \) |
| 13 | \( 1 + (9.14 + 7.67i)T + (29.3 + 166. i)T^{2} \) |
| 17 | \( 1 + (-2.93 - 1.69i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (10.2 + 17.7i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (0.678 + 0.119i)T + (497. + 180. i)T^{2} \) |
| 29 | \( 1 + (34.6 + 41.3i)T + (-146. + 828. i)T^{2} \) |
| 31 | \( 1 + (7.33 - 41.6i)T + (-903. - 328. i)T^{2} \) |
| 37 | \( 1 + (-8.24 + 14.2i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (26.0 - 30.9i)T + (-291. - 1.65e3i)T^{2} \) |
| 43 | \( 1 + (-66.4 - 24.1i)T + (1.41e3 + 1.18e3i)T^{2} \) |
| 47 | \( 1 + (-62.9 + 11.1i)T + (2.07e3 - 755. i)T^{2} \) |
| 53 | \( 1 - 17.6iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (-19.1 - 52.5i)T + (-2.66e3 + 2.23e3i)T^{2} \) |
| 61 | \( 1 + (-11.5 - 65.3i)T + (-3.49e3 + 1.27e3i)T^{2} \) |
| 67 | \( 1 + (17.2 + 14.5i)T + (779. + 4.42e3i)T^{2} \) |
| 71 | \( 1 + (-43.0 - 24.8i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (45.2 + 78.3i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (57.2 - 48.0i)T + (1.08e3 - 6.14e3i)T^{2} \) |
| 83 | \( 1 + (8.15 + 9.72i)T + (-1.19e3 + 6.78e3i)T^{2} \) |
| 89 | \( 1 + (58.5 - 33.7i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (44.2 + 16.1i)T + (7.20e3 + 6.04e3i)T^{2} \) |
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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
−12.18933083961381436353899215681, −11.19020581058143470008135913454, −10.42389946778437616373819390418, −9.188953705768490196262881481696, −8.401916118546248209069040836032, −7.20715311049918776820453412128, −5.78543043348391686094055016192, −4.19138451837824101308595872813, −2.79473386962015725592589857433, −0.69734514350760683760187242685,
1.98291258510197755502435016193, 4.14426758302769465088847771003, 5.49525376172092921061713447255, 6.88219505312601214248668062636, 7.53645165039085556541674565209, 9.001026907993956701075080596101, 9.695015345505278362651372710389, 10.78382545059356584404239665602, 12.03862810315271419011001221422, 12.81845578460306670527710840300
