| L(s) = 1 | + (−1.41 + 0.0389i)2-s + (0.788 + 1.02i)3-s + (1.99 − 0.110i)4-s + (1.15 − 1.51i)5-s + (−1.15 − 1.42i)6-s + (0.624 − 2.57i)7-s + (−2.81 + 0.233i)8-s + (0.342 − 1.27i)9-s + (−1.58 + 2.18i)10-s + (0.664 − 5.04i)11-s + (1.68 + 1.96i)12-s + (−2.64 + 6.37i)13-s + (−0.782 + 3.65i)14-s + 2.46·15-s + (3.97 − 0.439i)16-s + (0.520 − 0.901i)17-s + ⋯ |
| L(s) = 1 | + (−0.999 + 0.0275i)2-s + (0.454 + 0.592i)3-s + (0.998 − 0.0550i)4-s + (0.518 − 0.675i)5-s + (−0.471 − 0.580i)6-s + (0.236 − 0.971i)7-s + (−0.996 + 0.0825i)8-s + (0.114 − 0.426i)9-s + (−0.499 + 0.689i)10-s + (0.200 − 1.52i)11-s + (0.486 + 0.566i)12-s + (−0.732 + 1.76i)13-s + (−0.209 + 0.977i)14-s + 0.636·15-s + (0.993 − 0.109i)16-s + (0.126 − 0.218i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.908 + 0.418i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.908 + 0.418i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.990487 - 0.217330i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.990487 - 0.217330i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.41 - 0.0389i)T \) |
| 7 | \( 1 + (-0.624 + 2.57i)T \) |
| good | 3 | \( 1 + (-0.788 - 1.02i)T + (-0.776 + 2.89i)T^{2} \) |
| 5 | \( 1 + (-1.15 + 1.51i)T + (-1.29 - 4.82i)T^{2} \) |
| 11 | \( 1 + (-0.664 + 5.04i)T + (-10.6 - 2.84i)T^{2} \) |
| 13 | \( 1 + (2.64 - 6.37i)T + (-9.19 - 9.19i)T^{2} \) |
| 17 | \( 1 + (-0.520 + 0.901i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.185 + 1.41i)T + (-18.3 + 4.91i)T^{2} \) |
| 23 | \( 1 + (-3.16 - 0.846i)T + (19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (1.14 - 2.76i)T + (-20.5 - 20.5i)T^{2} \) |
| 31 | \( 1 + (1.53 - 2.66i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-7.90 - 6.06i)T + (9.57 + 35.7i)T^{2} \) |
| 41 | \( 1 + (-0.859 + 0.859i)T - 41iT^{2} \) |
| 43 | \( 1 + (5.93 - 2.45i)T + (30.4 - 30.4i)T^{2} \) |
| 47 | \( 1 + (1.27 - 0.738i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (8.48 + 1.11i)T + (51.1 + 13.7i)T^{2} \) |
| 59 | \( 1 + (1.46 - 11.1i)T + (-56.9 - 15.2i)T^{2} \) |
| 61 | \( 1 + (-1.68 - 12.8i)T + (-58.9 + 15.7i)T^{2} \) |
| 67 | \( 1 + (-7.53 + 5.78i)T + (17.3 - 64.7i)T^{2} \) |
| 71 | \( 1 + (8.22 + 8.22i)T + 71iT^{2} \) |
| 73 | \( 1 + (-0.468 - 1.74i)T + (-63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (6.23 + 10.8i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-5.52 - 2.28i)T + (58.6 + 58.6i)T^{2} \) |
| 89 | \( 1 + (-2.58 + 9.66i)T + (-77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + 8.78iT - 97T^{2} \) |
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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
−11.80595181545833014191348351078, −11.07317049036910838081913289156, −9.944124524973779027986600015467, −9.180713531096705762501660628852, −8.678153088480879538523550539752, −7.29329643547866423570544233833, −6.30044853863933496222572397553, −4.68500919374261012501740407172, −3.24575391888231233114789986896, −1.26413945640669481112676823941,
2.01667172558605068950743980630, 2.74505336217162902361817984043, 5.28584104693527240390988826995, 6.53859548561130820517035745706, 7.60187018416543435514738287657, 8.157156453601965449371304086276, 9.528640862653045083159098585303, 10.14795405413839324314997791074, 11.14354230582822074717906050684, 12.49797752259239773753176778763
