| L(s) = 1 | + (1.30 − 0.695i)3-s + (−2.50 − 3.05i)5-s + (−1.90 + 1.27i)7-s + (−0.456 + 0.683i)9-s + (−4.47 + 1.35i)11-s + (−2.41 − 1.98i)13-s + (−5.38 − 2.22i)15-s + (4.12 − 1.70i)17-s + (−0.371 − 3.77i)19-s + (−1.59 + 2.97i)21-s + (−1.98 + 0.395i)23-s + (−2.06 + 10.3i)25-s + (−0.552 + 5.61i)27-s + (0.870 − 2.86i)29-s + (−0.377 − 0.377i)31-s + ⋯ |
| L(s) = 1 | + (0.751 − 0.401i)3-s + (−1.11 − 1.36i)5-s + (−0.718 + 0.480i)7-s + (−0.152 + 0.227i)9-s + (−1.34 + 0.408i)11-s + (−0.671 − 0.550i)13-s + (−1.38 − 0.575i)15-s + (0.999 − 0.414i)17-s + (−0.0851 − 0.864i)19-s + (−0.347 + 0.649i)21-s + (−0.414 + 0.0825i)23-s + (−0.412 + 2.07i)25-s + (−0.106 + 1.08i)27-s + (0.161 − 0.532i)29-s + (−0.0677 − 0.0677i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 + 0.124i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.992 + 0.124i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0334971 - 0.536423i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0334971 - 0.536423i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + (-1.30 + 0.695i)T + (1.66 - 2.49i)T^{2} \) |
| 5 | \( 1 + (2.50 + 3.05i)T + (-0.975 + 4.90i)T^{2} \) |
| 7 | \( 1 + (1.90 - 1.27i)T + (2.67 - 6.46i)T^{2} \) |
| 11 | \( 1 + (4.47 - 1.35i)T + (9.14 - 6.11i)T^{2} \) |
| 13 | \( 1 + (2.41 + 1.98i)T + (2.53 + 12.7i)T^{2} \) |
| 17 | \( 1 + (-4.12 + 1.70i)T + (12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (0.371 + 3.77i)T + (-18.6 + 3.70i)T^{2} \) |
| 23 | \( 1 + (1.98 - 0.395i)T + (21.2 - 8.80i)T^{2} \) |
| 29 | \( 1 + (-0.870 + 2.86i)T + (-24.1 - 16.1i)T^{2} \) |
| 31 | \( 1 + (0.377 + 0.377i)T + 31iT^{2} \) |
| 37 | \( 1 + (-1.87 - 0.184i)T + (36.2 + 7.21i)T^{2} \) |
| 41 | \( 1 + (2.35 + 11.8i)T + (-37.8 + 15.6i)T^{2} \) |
| 43 | \( 1 + (2.04 + 1.09i)T + (23.8 + 35.7i)T^{2} \) |
| 47 | \( 1 + (0.807 + 1.95i)T + (-33.2 + 33.2i)T^{2} \) |
| 53 | \( 1 + (2.23 + 7.35i)T + (-44.0 + 29.4i)T^{2} \) |
| 59 | \( 1 + (-9.34 + 7.66i)T + (11.5 - 57.8i)T^{2} \) |
| 61 | \( 1 + (-4.03 - 7.55i)T + (-33.8 + 50.7i)T^{2} \) |
| 67 | \( 1 + (3.52 + 6.59i)T + (-37.2 + 55.7i)T^{2} \) |
| 71 | \( 1 + (5.81 + 8.70i)T + (-27.1 + 65.5i)T^{2} \) |
| 73 | \( 1 + (4.18 + 2.79i)T + (27.9 + 67.4i)T^{2} \) |
| 79 | \( 1 + (0.189 - 0.457i)T + (-55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (8.89 - 0.875i)T + (81.4 - 16.1i)T^{2} \) |
| 89 | \( 1 + (-5.75 - 1.14i)T + (82.2 + 34.0i)T^{2} \) |
| 97 | \( 1 + (-0.489 - 0.489i)T + 97iT^{2} \) |
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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
−10.32477214054216371546641162250, −9.368553005327968947884751244342, −8.518673917903147407060597909006, −7.85541759886938525872444330213, −7.29429737486698049975344134723, −5.47060396735940311709406739299, −4.86751721308132323404681741647, −3.41354057026054900398874385340, −2.36147492267076699037914881078, −0.27146312211112875878765436888,
2.80500832281000688332448179874, 3.32822400972591292223779703905, 4.24793666737306775751424528321, 5.92729683293709623793402407494, 6.96957685581394293407745519587, 7.79670178979577299609803890758, 8.407728849417333238533690885198, 9.975579126542819318965508371425, 10.13655228532202266942587862913, 11.22614948908393980066254789803
