| L(s) = 1 | + (−2.63 + 0.197i)2-s + (0.836 − 2.71i)3-s + (4.90 − 0.740i)4-s + (−0.744 + 2.10i)5-s + (−1.66 + 7.30i)6-s + (−1.48 + 2.18i)7-s + (−7.63 + 1.74i)8-s + (−4.17 − 2.84i)9-s + (1.54 − 5.69i)10-s + (−3.48 + 2.37i)11-s + (2.10 − 13.9i)12-s + (−1.37 + 2.84i)13-s + (3.48 − 6.04i)14-s + (5.09 + 3.78i)15-s + (10.2 − 3.16i)16-s + (−3.34 + 1.31i)17-s + ⋯ |
| L(s) = 1 | + (−1.86 + 0.139i)2-s + (0.483 − 1.56i)3-s + (2.45 − 0.370i)4-s + (−0.332 + 0.942i)5-s + (−0.680 + 2.98i)6-s + (−0.562 + 0.826i)7-s + (−2.69 + 0.615i)8-s + (−1.39 − 0.949i)9-s + (0.487 − 1.80i)10-s + (−1.05 + 0.716i)11-s + (0.606 − 4.02i)12-s + (−0.380 + 0.790i)13-s + (0.932 − 1.61i)14-s + (1.31 + 0.976i)15-s + (2.56 − 0.790i)16-s + (−0.812 + 0.318i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.318 - 0.947i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.318 - 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.116512 + 0.162096i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.116512 + 0.162096i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (0.744 - 2.10i)T \) |
| 7 | \( 1 + (1.48 - 2.18i)T \) |
| good | 2 | \( 1 + (2.63 - 0.197i)T + (1.97 - 0.298i)T^{2} \) |
| 3 | \( 1 + (-0.836 + 2.71i)T + (-2.47 - 1.68i)T^{2} \) |
| 11 | \( 1 + (3.48 - 2.37i)T + (4.01 - 10.2i)T^{2} \) |
| 13 | \( 1 + (1.37 - 2.84i)T + (-8.10 - 10.1i)T^{2} \) |
| 17 | \( 1 + (3.34 - 1.31i)T + (12.4 - 11.5i)T^{2} \) |
| 19 | \( 1 + (1.20 + 2.08i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.05 + 0.414i)T + (16.8 + 15.6i)T^{2} \) |
| 29 | \( 1 + (1.88 - 2.35i)T + (-6.45 - 28.2i)T^{2} \) |
| 31 | \( 1 + (1.03 - 1.78i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (0.373 - 2.47i)T + (-35.3 - 10.9i)T^{2} \) |
| 41 | \( 1 + (-0.0815 - 0.357i)T + (-36.9 + 17.7i)T^{2} \) |
| 43 | \( 1 + (-8.10 - 1.84i)T + (38.7 + 18.6i)T^{2} \) |
| 47 | \( 1 + (-8.75 + 0.656i)T + (46.4 - 7.00i)T^{2} \) |
| 53 | \( 1 + (0.852 + 5.65i)T + (-50.6 + 15.6i)T^{2} \) |
| 59 | \( 1 + (-1.58 - 1.47i)T + (4.40 + 58.8i)T^{2} \) |
| 61 | \( 1 + (4.98 + 0.751i)T + (58.2 + 17.9i)T^{2} \) |
| 67 | \( 1 + (0.295 + 0.170i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-4.70 - 5.90i)T + (-15.7 + 69.2i)T^{2} \) |
| 73 | \( 1 + (8.36 + 0.627i)T + (72.1 + 10.8i)T^{2} \) |
| 79 | \( 1 + (-6.16 - 10.6i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (5.97 + 12.4i)T + (-51.7 + 64.8i)T^{2} \) |
| 89 | \( 1 + (13.4 + 9.19i)T + (32.5 + 82.8i)T^{2} \) |
| 97 | \( 1 + 3.84iT - 97T^{2} \) |
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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
−12.16619698831240938832334983471, −11.30815225589855374853646962081, −10.29793876584508792051390668686, −9.203511515034910338238366759508, −8.405636715818958260026788473273, −7.44183370659451847406138132048, −6.95282304162093442839871426211, −6.14539693551385560137978802395, −2.65822238980133772647596099535, −2.09882886479792288547647597599,
0.24544162540958090507190407152, 2.79013482865768335981649221186, 4.09969283637489179181410849578, 5.66784004632475723650460096706, 7.50840501361112127409424450657, 8.230459839694108652724978758180, 9.099668719231439485159237574751, 9.727754888092981946626367688058, 10.59984754526609793107199305830, 11.00265315756305095029310239871
