| L(s) = 1 | + (−0.480 − 1.33i)2-s + (−0.758 + 0.988i)3-s + (−1.53 + 1.27i)4-s + (0.171 − 0.131i)5-s + (1.67 + 0.534i)6-s + (1.01 − 2.44i)7-s + (2.43 + 1.43i)8-s + (0.374 + 1.39i)9-s + (−0.256 − 0.164i)10-s + (2.10 − 0.276i)11-s + (−0.0961 − 2.49i)12-s + (3.53 − 1.46i)13-s + (−3.73 − 0.176i)14-s + 0.268i·15-s + (0.734 − 3.93i)16-s + (6.24 − 3.60i)17-s + ⋯ |
| L(s) = 1 | + (−0.339 − 0.940i)2-s + (−0.438 + 0.570i)3-s + (−0.769 + 0.638i)4-s + (0.0765 − 0.0587i)5-s + (0.685 + 0.218i)6-s + (0.383 − 0.923i)7-s + (0.862 + 0.506i)8-s + (0.124 + 0.465i)9-s + (−0.0812 − 0.0520i)10-s + (0.634 − 0.0834i)11-s + (−0.0277 − 0.719i)12-s + (0.980 − 0.406i)13-s + (−0.998 − 0.0471i)14-s + 0.0694i·15-s + (0.183 − 0.982i)16-s + (1.51 − 0.874i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.726 + 0.687i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.726 + 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.888285 - 0.353500i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.888285 - 0.353500i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.480 + 1.33i)T \) |
| 7 | \( 1 + (-1.01 + 2.44i)T \) |
| good | 3 | \( 1 + (0.758 - 0.988i)T + (-0.776 - 2.89i)T^{2} \) |
| 5 | \( 1 + (-0.171 + 0.131i)T + (1.29 - 4.82i)T^{2} \) |
| 11 | \( 1 + (-2.10 + 0.276i)T + (10.6 - 2.84i)T^{2} \) |
| 13 | \( 1 + (-3.53 + 1.46i)T + (9.19 - 9.19i)T^{2} \) |
| 17 | \( 1 + (-6.24 + 3.60i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.462 - 3.51i)T + (-18.3 - 4.91i)T^{2} \) |
| 23 | \( 1 + (1.34 + 5.02i)T + (-19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (-2.56 - 6.18i)T + (-20.5 + 20.5i)T^{2} \) |
| 31 | \( 1 + (2.46 + 4.27i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (2.06 - 1.58i)T + (9.57 - 35.7i)T^{2} \) |
| 41 | \( 1 + (-4.31 + 4.31i)T - 41iT^{2} \) |
| 43 | \( 1 + (4.15 - 10.0i)T + (-30.4 - 30.4i)T^{2} \) |
| 47 | \( 1 + (-4.71 - 2.71i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-6.20 + 0.817i)T + (51.1 - 13.7i)T^{2} \) |
| 59 | \( 1 + (0.990 + 7.52i)T + (-56.9 + 15.2i)T^{2} \) |
| 61 | \( 1 + (0.817 + 0.107i)T + (58.9 + 15.7i)T^{2} \) |
| 67 | \( 1 + (2.72 - 3.55i)T + (-17.3 - 64.7i)T^{2} \) |
| 71 | \( 1 + (2.53 + 2.53i)T + 71iT^{2} \) |
| 73 | \( 1 + (11.7 + 3.13i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (2.02 + 1.16i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (1.12 - 0.467i)T + (58.6 - 58.6i)T^{2} \) |
| 89 | \( 1 + (16.0 - 4.29i)T + (77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 - 10.3T + 97T^{2} \) |
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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
−11.85516562787106030402219606295, −10.99688743761603753834883276347, −10.38010944707407364852358430237, −9.583765192847917152805455501481, −8.314540549217974292509372008164, −7.41154190951474639098431610716, −5.59330929475558512488808857999, −4.43066247131565615887718523664, −3.42807218868649667568348229854, −1.30149533186305550350952013544,
1.41404452872849767656102253488, 3.96982462148214493592145036715, 5.62575548419847900032519483642, 6.17293406913801435503714586905, 7.23936939231585172860687083101, 8.384247921090486739112272283531, 9.164049563674214346769164482595, 10.24715449379402839919671923060, 11.61354243322158934862837210109, 12.29103136102996264996291438892
