| L(s) = 1 | + (−0.654 − 0.755i)2-s + (2.64 + 1.69i)3-s + (−0.142 + 0.989i)4-s + (0.940 − 2.05i)5-s + (−0.447 − 3.11i)6-s + (0.959 + 0.281i)7-s + (0.841 − 0.540i)8-s + (2.85 + 6.26i)9-s + (−2.17 + 0.637i)10-s + (2.81 − 3.24i)11-s + (−2.05 + 2.37i)12-s + (−6.57 + 1.93i)13-s + (−0.415 − 0.909i)14-s + (5.98 − 3.84i)15-s + (−0.959 − 0.281i)16-s + (−0.656 − 4.56i)17-s + ⋯ |
| L(s) = 1 | + (−0.463 − 0.534i)2-s + (1.52 + 0.981i)3-s + (−0.0711 + 0.494i)4-s + (0.420 − 0.921i)5-s + (−0.182 − 1.27i)6-s + (0.362 + 0.106i)7-s + (0.297 − 0.191i)8-s + (0.953 + 2.08i)9-s + (−0.687 + 0.201i)10-s + (0.847 − 0.978i)11-s + (−0.594 + 0.685i)12-s + (−1.82 + 0.535i)13-s + (−0.111 − 0.243i)14-s + (1.54 − 0.993i)15-s + (−0.239 − 0.0704i)16-s + (−0.159 − 1.10i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0683i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0683i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.78010 - 0.0609067i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.78010 - 0.0609067i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.654 + 0.755i)T \) |
| 7 | \( 1 + (-0.959 - 0.281i)T \) |
| 23 | \( 1 + (4.50 - 1.63i)T \) |
| good | 3 | \( 1 + (-2.64 - 1.69i)T + (1.24 + 2.72i)T^{2} \) |
| 5 | \( 1 + (-0.940 + 2.05i)T + (-3.27 - 3.77i)T^{2} \) |
| 11 | \( 1 + (-2.81 + 3.24i)T + (-1.56 - 10.8i)T^{2} \) |
| 13 | \( 1 + (6.57 - 1.93i)T + (10.9 - 7.02i)T^{2} \) |
| 17 | \( 1 + (0.656 + 4.56i)T + (-16.3 + 4.78i)T^{2} \) |
| 19 | \( 1 + (0.299 - 2.08i)T + (-18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (-1.33 - 9.28i)T + (-27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (-5.45 + 3.50i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + (2.83 + 6.20i)T + (-24.2 + 27.9i)T^{2} \) |
| 41 | \( 1 + (0.00243 - 0.00533i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (4.17 + 2.68i)T + (17.8 + 39.1i)T^{2} \) |
| 47 | \( 1 + 5.83T + 47T^{2} \) |
| 53 | \( 1 + (-2.03 - 0.598i)T + (44.5 + 28.6i)T^{2} \) |
| 59 | \( 1 + (-1.91 + 0.561i)T + (49.6 - 31.8i)T^{2} \) |
| 61 | \( 1 + (5.19 - 3.33i)T + (25.3 - 55.4i)T^{2} \) |
| 67 | \( 1 + (-1.90 - 2.20i)T + (-9.53 + 66.3i)T^{2} \) |
| 71 | \( 1 + (1.52 + 1.75i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (-1.61 + 11.2i)T + (-70.0 - 20.5i)T^{2} \) |
| 79 | \( 1 + (15.6 - 4.59i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (1.53 + 3.35i)T + (-54.3 + 62.7i)T^{2} \) |
| 89 | \( 1 + (-0.746 - 0.479i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (3.58 - 7.84i)T + (-63.5 - 73.3i)T^{2} \) |
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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
−11.56122059334895831426634930317, −10.26994820888813329643329261904, −9.511890699228653102771289813940, −9.026545637367356016586374381919, −8.328431373237070401054283573084, −7.26823799145383409769588728187, −5.16008252183013546823671280393, −4.29898178374696001535866996148, −3.06667061813298245239369972953, −1.86585345666221665134010633994,
1.84656815857084976131889402879, 2.73891323860478448030493986498, 4.42547906833816122689327656071, 6.40535562736775723709993474890, 6.96826600735373747994924074856, 7.83513418420079886998950941434, 8.512341080096811797473450266630, 9.871716442037932270823654963011, 10.02002018125279759476511489330, 11.85047945905533519584674641315
