| 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
−11.22614948908393980066254789803, −10.13655228532202266942587862913, −9.975579126542819318965508371425, −8.407728849417333238533690885198, −7.79670178979577299609803890758, −6.96957685581394293407745519587, −5.92729683293709623793402407494, −4.24793666737306775751424528321, −3.32822400972591292223779703905, −2.80500832281000688332448179874,
0.27146312211112875878765436888, 2.36147492267076699037914881078, 3.41354057026054900398874385340, 4.86751721308132323404681741647, 5.47060396735940311709406739299, 7.29429737486698049975344134723, 7.85541759886938525872444330213, 8.518673917903147407060597909006, 9.368553005327968947884751244342, 10.32477214054216371546641162250
