| L(s) = 1 | + (−3.78 + 0.180i)2-s + (2.93 + 4.29i)3-s + (6.30 − 0.602i)4-s + (−11.8 − 11.2i)5-s + (−11.8 − 15.6i)6-s + (3.18 + 16.5i)7-s + (6.24 − 0.898i)8-s + (−9.82 + 25.1i)9-s + (46.7 + 40.4i)10-s + (−4.52 + 2.33i)11-s + (21.0 + 25.2i)12-s + (−31.9 − 6.15i)13-s + (−15.0 − 61.9i)14-s + (13.7 − 83.7i)15-s + (−73.2 + 14.1i)16-s + (−33.5 − 73.4i)17-s + ⋯ |
| L(s) = 1 | + (−1.33 + 0.0636i)2-s + (0.563 + 0.825i)3-s + (0.788 − 0.0752i)4-s + (−1.05 − 1.00i)5-s + (−0.806 − 1.06i)6-s + (0.172 + 0.892i)7-s + (0.276 − 0.0396i)8-s + (−0.363 + 0.931i)9-s + (1.47 + 1.27i)10-s + (−0.124 + 0.0639i)11-s + (0.506 + 0.608i)12-s + (−0.681 − 0.131i)13-s + (−0.286 − 1.18i)14-s + (0.236 − 1.44i)15-s + (−1.14 + 0.220i)16-s + (−0.478 − 1.04i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.665 + 0.746i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.665 + 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.495080 - 0.221726i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.495080 - 0.221726i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-2.93 - 4.29i)T \) |
| 23 | \( 1 + (-88.5 + 65.7i)T \) |
| good | 2 | \( 1 + (3.78 - 0.180i)T + (7.96 - 0.760i)T^{2} \) |
| 5 | \( 1 + (11.8 + 11.2i)T + (5.94 + 124. i)T^{2} \) |
| 7 | \( 1 + (-3.18 - 16.5i)T + (-318. + 127. i)T^{2} \) |
| 11 | \( 1 + (4.52 - 2.33i)T + (772. - 1.08e3i)T^{2} \) |
| 13 | \( 1 + (31.9 + 6.15i)T + (2.03e3 + 816. i)T^{2} \) |
| 17 | \( 1 + (33.5 + 73.4i)T + (-3.21e3 + 3.71e3i)T^{2} \) |
| 19 | \( 1 + (-24.4 - 11.1i)T + (4.49e3 + 5.18e3i)T^{2} \) |
| 29 | \( 1 + (2.38 - 24.9i)T + (-2.39e4 - 4.61e3i)T^{2} \) |
| 31 | \( 1 + (-134. + 105. i)T + (7.02e3 - 2.89e4i)T^{2} \) |
| 37 | \( 1 + (-29.7 + 101. i)T + (-4.26e4 - 2.73e4i)T^{2} \) |
| 41 | \( 1 + (-242. + 254. i)T + (-3.27e3 - 6.88e4i)T^{2} \) |
| 43 | \( 1 + (-159. + 203. i)T + (-1.87e4 - 7.72e4i)T^{2} \) |
| 47 | \( 1 + (-177. - 102. i)T + (5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (28.5 + 32.8i)T + (-2.11e4 + 1.47e5i)T^{2} \) |
| 59 | \( 1 + (-140. + 730. i)T + (-1.90e5 - 7.63e4i)T^{2} \) |
| 61 | \( 1 + (148. - 369. i)T + (-1.64e5 - 1.56e5i)T^{2} \) |
| 67 | \( 1 + (196. - 381. i)T + (-1.74e5 - 2.44e5i)T^{2} \) |
| 71 | \( 1 + (607. + 944. i)T + (-1.48e5 + 3.25e5i)T^{2} \) |
| 73 | \( 1 + (-295. + 646. i)T + (-2.54e5 - 2.93e5i)T^{2} \) |
| 79 | \( 1 + (-1.01e3 + 350. i)T + (3.87e5 - 3.04e5i)T^{2} \) |
| 83 | \( 1 + (997. - 951. i)T + (2.72e4 - 5.71e5i)T^{2} \) |
| 89 | \( 1 + (88.4 - 615. i)T + (-6.76e5 - 1.98e5i)T^{2} \) |
| 97 | \( 1 + (-235. - 57.1i)T + (8.11e5 + 4.18e5i)T^{2} \) |
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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
−11.56540747329931646584996757121, −10.60262563276173110227805859386, −9.354120139161042997543737367655, −8.992993028177793877024603925715, −8.136599568233997239664694552885, −7.37791742507710450311087268767, −5.16913637718991005273238622842, −4.31302855821304779334708171503, −2.48268792517138432054811299175, −0.41031898412990088913690106149,
1.09403435418999106810250712253, 2.76886754153844163614373536746, 4.18367530575244772457300885701, 6.63476944682162529798057411329, 7.42857585129255643309784256960, 7.87563610477699074449108274433, 8.916141765508471807698070344567, 10.10049016181972755244015568363, 10.97206678365203243174298946912, 11.70815505843940203546924081350
