| L(s) = 1 | + (0.148 + 1.40i)2-s + (−1.23 + 1.60i)3-s + (−1.95 + 0.416i)4-s + (0.591 − 0.453i)5-s + (−2.43 − 1.49i)6-s + (−2.18 + 1.49i)7-s + (−0.876 − 2.68i)8-s + (−0.280 − 1.04i)9-s + (0.725 + 0.764i)10-s + (−3.27 + 0.431i)11-s + (1.73 − 3.64i)12-s + (0.237 − 0.0983i)13-s + (−2.42 − 2.85i)14-s + 1.50i·15-s + (3.65 − 1.63i)16-s + (0.237 − 0.137i)17-s + ⋯ |
| L(s) = 1 | + (0.104 + 0.994i)2-s + (−0.710 + 0.925i)3-s + (−0.978 + 0.208i)4-s + (0.264 − 0.202i)5-s + (−0.995 − 0.609i)6-s + (−0.826 + 0.563i)7-s + (−0.309 − 0.950i)8-s + (−0.0936 − 0.349i)9-s + (0.229 + 0.241i)10-s + (−0.987 + 0.129i)11-s + (0.501 − 1.05i)12-s + (0.0658 − 0.0272i)13-s + (−0.646 − 0.762i)14-s + 0.389i·15-s + (0.913 − 0.407i)16-s + (0.0576 − 0.0333i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.826 + 0.562i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.826 + 0.562i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.170814 - 0.554524i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.170814 - 0.554524i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.148 - 1.40i)T \) |
| 7 | \( 1 + (2.18 - 1.49i)T \) |
| good | 3 | \( 1 + (1.23 - 1.60i)T + (-0.776 - 2.89i)T^{2} \) |
| 5 | \( 1 + (-0.591 + 0.453i)T + (1.29 - 4.82i)T^{2} \) |
| 11 | \( 1 + (3.27 - 0.431i)T + (10.6 - 2.84i)T^{2} \) |
| 13 | \( 1 + (-0.237 + 0.0983i)T + (9.19 - 9.19i)T^{2} \) |
| 17 | \( 1 + (-0.237 + 0.137i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.413 + 3.14i)T + (-18.3 - 4.91i)T^{2} \) |
| 23 | \( 1 + (-1.45 - 5.44i)T + (-19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (-0.801 - 1.93i)T + (-20.5 + 20.5i)T^{2} \) |
| 31 | \( 1 + (-1.52 - 2.63i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (6.47 - 4.96i)T + (9.57 - 35.7i)T^{2} \) |
| 41 | \( 1 + (2.41 - 2.41i)T - 41iT^{2} \) |
| 43 | \( 1 + (3.81 - 9.21i)T + (-30.4 - 30.4i)T^{2} \) |
| 47 | \( 1 + (-8.91 - 5.14i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-7.96 + 1.04i)T + (51.1 - 13.7i)T^{2} \) |
| 59 | \( 1 + (1.51 + 11.5i)T + (-56.9 + 15.2i)T^{2} \) |
| 61 | \( 1 + (6.27 + 0.825i)T + (58.9 + 15.7i)T^{2} \) |
| 67 | \( 1 + (-0.961 + 1.25i)T + (-17.3 - 64.7i)T^{2} \) |
| 71 | \( 1 + (1.35 + 1.35i)T + 71iT^{2} \) |
| 73 | \( 1 + (0.883 + 0.236i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (-12.0 - 6.93i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-15.5 + 6.42i)T + (58.6 - 58.6i)T^{2} \) |
| 89 | \( 1 + (-10.2 + 2.73i)T + (77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + 0.0930T + 97T^{2} \) |
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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
−13.07250133247019988426914503396, −11.94648036761056499995916202535, −10.66866859080369939918047058972, −9.738173581068957022171618182581, −9.085910737750506856575932428578, −7.76771922778774018563742957676, −6.52452596485447907244340070927, −5.42274273643802502591585620915, −4.92319626231464045650962433038, −3.33311327982142434770813171069,
0.49960817411879152392791171893, 2.34797092435282714450498835060, 3.84716735013145534559147300559, 5.45600621427006256568197292512, 6.40870121506494699134577782927, 7.58196355742539917438416356714, 8.891183211687842304604472197060, 10.31464640803985929687651468539, 10.52705232361258920340272384966, 11.99240801899723095735247018711
