| L(s) = 1 | + (0.642 + 1.26i)2-s + (−0.270 − 1.71i)3-s + (−1.17 + 1.61i)4-s + (−3.75 + 3.30i)5-s + (1.98 − 1.43i)6-s + (−5.04 + 5.04i)7-s + (−2.79 − 0.442i)8-s + (−2.85 + 0.927i)9-s + (−6.57 − 2.61i)10-s + (−1.94 + 5.99i)11-s + (3.08 + 1.57i)12-s + (−0.134 + 0.264i)13-s + (−9.60 − 3.12i)14-s + (6.66 + 5.53i)15-s + (−1.23 − 3.80i)16-s + (−3.88 + 24.5i)17-s + ⋯ |
| L(s) = 1 | + (0.321 + 0.630i)2-s + (−0.0903 − 0.570i)3-s + (−0.293 + 0.404i)4-s + (−0.751 + 0.660i)5-s + (0.330 − 0.239i)6-s + (−0.721 + 0.721i)7-s + (−0.349 − 0.0553i)8-s + (−0.317 + 0.103i)9-s + (−0.657 − 0.261i)10-s + (−0.176 + 0.544i)11-s + (0.257 + 0.131i)12-s + (−0.0103 + 0.0203i)13-s + (−0.686 − 0.222i)14-s + (0.444 + 0.368i)15-s + (−0.0772 − 0.237i)16-s + (−0.228 + 1.44i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.874 - 0.484i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.874 - 0.484i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.212179 + 0.821183i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.212179 + 0.821183i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.642 - 1.26i)T \) |
| 3 | \( 1 + (0.270 + 1.71i)T \) |
| 5 | \( 1 + (3.75 - 3.30i)T \) |
| good | 7 | \( 1 + (5.04 - 5.04i)T - 49iT^{2} \) |
| 11 | \( 1 + (1.94 - 5.99i)T + (-97.8 - 71.1i)T^{2} \) |
| 13 | \( 1 + (0.134 - 0.264i)T + (-99.3 - 136. i)T^{2} \) |
| 17 | \( 1 + (3.88 - 24.5i)T + (-274. - 89.3i)T^{2} \) |
| 19 | \( 1 + (3.92 + 5.40i)T + (-111. + 343. i)T^{2} \) |
| 23 | \( 1 + (-0.779 + 0.397i)T + (310. - 427. i)T^{2} \) |
| 29 | \( 1 + (-0.149 + 0.205i)T + (-259. - 799. i)T^{2} \) |
| 31 | \( 1 + (-31.6 + 23.0i)T + (296. - 913. i)T^{2} \) |
| 37 | \( 1 + (10.5 + 5.36i)T + (804. + 1.10e3i)T^{2} \) |
| 41 | \( 1 + (-13.4 - 41.3i)T + (-1.35e3 + 988. i)T^{2} \) |
| 43 | \( 1 + (9.20 + 9.20i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (-27.3 + 4.32i)T + (2.10e3 - 682. i)T^{2} \) |
| 53 | \( 1 + (-16.2 - 102. i)T + (-2.67e3 + 868. i)T^{2} \) |
| 59 | \( 1 + (-19.5 + 6.35i)T + (2.81e3 - 2.04e3i)T^{2} \) |
| 61 | \( 1 + (-24.2 + 74.7i)T + (-3.01e3 - 2.18e3i)T^{2} \) |
| 67 | \( 1 + (14.1 - 89.0i)T + (-4.26e3 - 1.38e3i)T^{2} \) |
| 71 | \( 1 + (-70.3 - 51.1i)T + (1.55e3 + 4.79e3i)T^{2} \) |
| 73 | \( 1 + (126. - 64.5i)T + (3.13e3 - 4.31e3i)T^{2} \) |
| 79 | \( 1 + (-68.2 + 93.8i)T + (-1.92e3 - 5.93e3i)T^{2} \) |
| 83 | \( 1 + (87.8 + 13.9i)T + (6.55e3 + 2.12e3i)T^{2} \) |
| 89 | \( 1 + (141. + 46.0i)T + (6.40e3 + 4.65e3i)T^{2} \) |
| 97 | \( 1 + (56.9 - 9.02i)T + (8.94e3 - 2.90e3i)T^{2} \) |
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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
−13.04868714909808750654754037262, −12.43279389723432428141972210813, −11.45604142872513596625152054843, −10.17650120499188852729663644204, −8.744783962093152233988065350058, −7.76898291674040975543197051184, −6.71881666824601388482853968820, −5.91376296091282467488531039568, −4.20012524893392944854305834293, −2.70334564390253342175254967138,
0.48975141988223531352117961446, 3.16298061032108540581770115623, 4.23147903583388293792576443492, 5.32645515504951965082244175351, 6.94865497185856171114337803356, 8.413505659199993296504134836599, 9.461529420232078302304694581699, 10.42528247325519627640730252323, 11.41056356881433102337663179529, 12.24349001131059602820385696756
