| L(s) = 1 | + (0.342 − 0.939i)2-s + (1.29 + 1.15i)3-s + (−0.766 − 0.642i)4-s + (0.588 − 3.34i)5-s + (1.52 − 0.819i)6-s + (0.0171 + 2.64i)7-s + (−0.866 + 0.500i)8-s + (0.338 + 2.98i)9-s + (−2.93 − 1.69i)10-s + (5.82 − 1.02i)11-s + (−0.248 − 1.71i)12-s + (−1.71 − 4.72i)13-s + (2.49 + 0.888i)14-s + (4.61 − 3.63i)15-s + (0.173 + 0.984i)16-s + (0.893 − 1.54i)17-s + ⋯ |
| L(s) = 1 | + (0.241 − 0.664i)2-s + (0.745 + 0.666i)3-s + (−0.383 − 0.321i)4-s + (0.263 − 1.49i)5-s + (0.622 − 0.334i)6-s + (0.00647 + 0.999i)7-s + (−0.306 + 0.176i)8-s + (0.112 + 0.993i)9-s + (−0.928 − 0.536i)10-s + (1.75 − 0.309i)11-s + (−0.0716 − 0.494i)12-s + (−0.476 − 1.30i)13-s + (0.666 + 0.237i)14-s + (1.19 − 0.938i)15-s + (0.0434 + 0.246i)16-s + (0.216 − 0.375i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.621 + 0.783i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.621 + 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.79361 - 0.866850i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.79361 - 0.866850i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.342 + 0.939i)T \) |
| 3 | \( 1 + (-1.29 - 1.15i)T \) |
| 7 | \( 1 + (-0.0171 - 2.64i)T \) |
| good | 5 | \( 1 + (-0.588 + 3.34i)T + (-4.69 - 1.71i)T^{2} \) |
| 11 | \( 1 + (-5.82 + 1.02i)T + (10.3 - 3.76i)T^{2} \) |
| 13 | \( 1 + (1.71 + 4.72i)T + (-9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-0.893 + 1.54i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.39 - 0.802i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.97 - 3.54i)T + (-3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (0.134 - 0.369i)T + (-22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (-5.48 + 6.54i)T + (-5.38 - 30.5i)T^{2} \) |
| 37 | \( 1 + (3.02 - 5.24i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (4.59 - 1.67i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (-1.47 - 8.34i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (4.18 - 3.50i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + 3.16iT - 53T^{2} \) |
| 59 | \( 1 + (0.911 - 5.17i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (-2.75 - 3.28i)T + (-10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (10.1 - 3.68i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (-8.84 - 5.10i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-1.36 + 0.786i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (12.3 + 4.47i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-4.62 - 1.68i)T + (63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + (-0.381 - 0.661i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (6.77 - 1.19i)T + (91.1 - 33.1i)T^{2} \) |
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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
−11.47825075257102004386540866051, −9.921562161080982781284599354884, −9.514821509236825036882622736906, −8.680622141424288351925376567312, −8.073150378821621765761120001940, −5.98109999671978215027619007112, −5.10134124522517705287464690888, −4.18704516486603910746882780004, −2.95589251768542354216035267820, −1.48466127062929577228911326669,
1.91499852823149317884352890146, 3.47244084873978781804169232872, 4.25505032169405983042574181639, 6.47303465565432532178164892776, 6.70401196516778202767542205812, 7.33663739735312035399715732549, 8.591431683769477505602824419764, 9.551187460041970587716496312656, 10.44082180593717771984143145876, 11.68093814596861237571899382300
