| L(s) = 1 | + (−0.628 + 1.26i)2-s + (−0.274 + 2.79i)3-s + (−1.20 − 1.59i)4-s + (2.09 − 1.11i)5-s + (−3.36 − 2.10i)6-s + (0.195 + 0.980i)7-s + (2.77 − 0.529i)8-s + (−4.76 − 0.948i)9-s + (0.100 + 3.35i)10-s + (−2.39 + 2.92i)11-s + (4.77 − 2.93i)12-s + (−0.322 + 0.603i)13-s + (−1.36 − 0.369i)14-s + (2.54 + 6.15i)15-s + (−1.07 + 3.85i)16-s + (1.28 − 3.09i)17-s + ⋯ |
| L(s) = 1 | + (−0.444 + 0.895i)2-s + (−0.158 + 1.61i)3-s + (−0.604 − 0.796i)4-s + (0.936 − 0.500i)5-s + (−1.37 − 0.858i)6-s + (0.0737 + 0.370i)7-s + (0.982 − 0.187i)8-s + (−1.58 − 0.316i)9-s + (0.0318 + 1.06i)10-s + (−0.723 + 0.881i)11-s + (1.37 − 0.847i)12-s + (−0.0894 + 0.167i)13-s + (−0.364 − 0.0988i)14-s + (0.658 + 1.58i)15-s + (−0.269 + 0.963i)16-s + (0.310 − 0.749i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.661 + 0.750i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.661 + 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.332031 - 0.735357i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.332031 - 0.735357i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.628 - 1.26i)T \) |
| 7 | \( 1 + (-0.195 - 0.980i)T \) |
| good | 3 | \( 1 + (0.274 - 2.79i)T + (-2.94 - 0.585i)T^{2} \) |
| 5 | \( 1 + (-2.09 + 1.11i)T + (2.77 - 4.15i)T^{2} \) |
| 11 | \( 1 + (2.39 - 2.92i)T + (-2.14 - 10.7i)T^{2} \) |
| 13 | \( 1 + (0.322 - 0.603i)T + (-7.22 - 10.8i)T^{2} \) |
| 17 | \( 1 + (-1.28 + 3.09i)T + (-12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (-0.0142 - 0.0469i)T + (-15.7 + 10.5i)T^{2} \) |
| 23 | \( 1 + (7.59 - 5.07i)T + (8.80 - 21.2i)T^{2} \) |
| 29 | \( 1 + (0.672 - 0.551i)T + (5.65 - 28.4i)T^{2} \) |
| 31 | \( 1 + (2.61 - 2.61i)T - 31iT^{2} \) |
| 37 | \( 1 + (11.1 + 3.36i)T + (30.7 + 20.5i)T^{2} \) |
| 41 | \( 1 + (-5.26 - 7.87i)T + (-15.6 + 37.8i)T^{2} \) |
| 43 | \( 1 + (-0.175 - 1.78i)T + (-42.1 + 8.38i)T^{2} \) |
| 47 | \( 1 + (7.83 + 3.24i)T + (33.2 + 33.2i)T^{2} \) |
| 53 | \( 1 + (-5.42 - 4.45i)T + (10.3 + 51.9i)T^{2} \) |
| 59 | \( 1 + (1.54 + 2.88i)T + (-32.7 + 49.0i)T^{2} \) |
| 61 | \( 1 + (-8.28 - 0.815i)T + (59.8 + 11.9i)T^{2} \) |
| 67 | \( 1 + (-9.59 - 0.945i)T + (65.7 + 13.0i)T^{2} \) |
| 71 | \( 1 + (-3.49 + 0.695i)T + (65.5 - 27.1i)T^{2} \) |
| 73 | \( 1 + (-0.685 + 3.44i)T + (-67.4 - 27.9i)T^{2} \) |
| 79 | \( 1 + (-1.46 + 0.607i)T + (55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (-14.4 + 4.37i)T + (69.0 - 46.1i)T^{2} \) |
| 89 | \( 1 + (-1.96 - 1.31i)T + (34.0 + 82.2i)T^{2} \) |
| 97 | \( 1 + (5.50 - 5.50i)T - 97iT^{2} \) |
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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
−10.14110275526380697187896523242, −9.687153015453485282507736513877, −9.282026740202833138256735055568, −8.335401244775165573254341400867, −7.33203198425440021100148947850, −6.04454919181054669985710327747, −5.21898091007620171123075351277, −4.96733612887310744691438358311, −3.73916779665668160882033561767, −1.96563835342024302925667554359,
0.43435726701545284716049466035, 1.83617486603284257259846260628, 2.45877641155741452040298829094, 3.70648421167365328817325230104, 5.42383707061086283399836263996, 6.23812697834143924515854908551, 7.12335094781202054921146663132, 8.085806742976496946067748330186, 8.469591878645189284071628193102, 9.854026777554773171735752794232
