| L(s) = 1 | + (−0.173 − 0.984i)3-s + (1.00 − 0.844i)5-s + (−3.15 − 1.82i)7-s + (−0.939 + 0.342i)9-s + (−2.08 + 1.20i)11-s + (−1.01 − 0.179i)13-s + (−1.00 − 0.844i)15-s + (−0.154 − 0.0563i)17-s + (−4.09 + 1.48i)19-s + (−1.24 + 3.42i)21-s + (−1.55 + 1.84i)23-s + (−0.568 + 3.22i)25-s + (0.5 + 0.866i)27-s + (−0.705 − 1.93i)29-s + (0.920 − 1.59i)31-s + ⋯ |
| L(s) = 1 | + (−0.100 − 0.568i)3-s + (0.449 − 0.377i)5-s + (−1.19 − 0.689i)7-s + (−0.313 + 0.114i)9-s + (−0.627 + 0.362i)11-s + (−0.282 − 0.0498i)13-s + (−0.259 − 0.217i)15-s + (−0.0375 − 0.0136i)17-s + (−0.940 + 0.339i)19-s + (−0.272 + 0.748i)21-s + (−0.323 + 0.385i)23-s + (−0.113 + 0.645i)25-s + (0.0962 + 0.166i)27-s + (−0.131 − 0.360i)29-s + (0.165 − 0.286i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.890 - 0.454i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.890 - 0.454i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0500661 + 0.208247i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0500661 + 0.208247i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.173 + 0.984i)T \) |
| 19 | \( 1 + (4.09 - 1.48i)T \) |
| good | 5 | \( 1 + (-1.00 + 0.844i)T + (0.868 - 4.92i)T^{2} \) |
| 7 | \( 1 + (3.15 + 1.82i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (2.08 - 1.20i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (1.01 + 0.179i)T + (12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (0.154 + 0.0563i)T + (13.0 + 10.9i)T^{2} \) |
| 23 | \( 1 + (1.55 - 1.84i)T + (-3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (0.705 + 1.93i)T + (-22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (-0.920 + 1.59i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 2.17iT - 37T^{2} \) |
| 41 | \( 1 + (4.45 - 0.785i)T + (38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (0.789 + 0.940i)T + (-7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (0.0928 + 0.254i)T + (-36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + (-3.02 + 3.60i)T + (-9.20 - 52.1i)T^{2} \) |
| 59 | \( 1 + (7.95 + 2.89i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-1.18 - 0.996i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (-3.43 + 1.25i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (4.24 - 3.56i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (-0.335 - 1.90i)T + (-68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (1.63 + 9.27i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (15.2 + 8.81i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (16.9 + 2.98i)T + (83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (5.59 - 15.3i)T + (-74.3 - 62.3i)T^{2} \) |
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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
−9.819926249091950476811279093056, −8.787498198417034155220550906949, −7.80436135925124115133140219550, −7.01024948060044049840281682343, −6.23188755702099962429432109611, −5.36577063842812827991892709605, −4.17503162007149361947625921034, −3.00249671054231936167214564214, −1.75590626802291516050102284863, −0.094246628205369093347940660990,
2.37945419904969704001271041495, 3.11960551307043505270160680376, 4.34118459017397547018840236928, 5.50000638846212340974458427800, 6.19390006913328142592208501731, 6.94348824272816846589350543492, 8.291490215567223684432793432457, 9.010963676878085921467595164380, 9.863425680858491137122736668244, 10.38411971847576878194577925301
