| L(s) = 1 | + (−0.345 + 0.0760i)2-s + (0.912 − 1.47i)3-s + (−1.70 + 0.787i)4-s + (2.86 + 0.794i)5-s + (−0.203 + 0.577i)6-s + (0.822 − 0.779i)7-s + (1.09 − 0.829i)8-s + (−1.33 − 2.68i)9-s + (−1.04 − 0.0568i)10-s + (1.99 + 2.35i)11-s + (−0.394 + 3.22i)12-s + (−0.823 − 2.44i)13-s + (−0.224 + 0.331i)14-s + (3.78 − 3.48i)15-s + (2.11 − 2.48i)16-s + (0.933 − 0.985i)17-s + ⋯ |
| L(s) = 1 | + (−0.244 + 0.0537i)2-s + (0.527 − 0.849i)3-s + (−0.850 + 0.393i)4-s + (1.27 + 0.355i)5-s + (−0.0830 + 0.235i)6-s + (0.310 − 0.294i)7-s + (0.385 − 0.293i)8-s + (−0.444 − 0.895i)9-s + (−0.331 − 0.0179i)10-s + (0.602 + 0.709i)11-s + (−0.113 + 0.930i)12-s + (−0.228 − 0.677i)13-s + (−0.0600 + 0.0886i)14-s + (0.976 − 0.900i)15-s + (0.528 − 0.622i)16-s + (0.226 − 0.239i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.901 + 0.432i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.901 + 0.432i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.21239 - 0.276075i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.21239 - 0.276075i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.912 + 1.47i)T \) |
| 59 | \( 1 + (5.90 + 4.91i)T \) |
| good | 2 | \( 1 + (0.345 - 0.0760i)T + (1.81 - 0.839i)T^{2} \) |
| 5 | \( 1 + (-2.86 - 0.794i)T + (4.28 + 2.57i)T^{2} \) |
| 7 | \( 1 + (-0.822 + 0.779i)T + (0.378 - 6.98i)T^{2} \) |
| 11 | \( 1 + (-1.99 - 2.35i)T + (-1.77 + 10.8i)T^{2} \) |
| 13 | \( 1 + (0.823 + 2.44i)T + (-10.3 + 7.86i)T^{2} \) |
| 17 | \( 1 + (-0.933 + 0.985i)T + (-0.920 - 16.9i)T^{2} \) |
| 19 | \( 1 + (-0.313 - 0.786i)T + (-13.7 + 13.0i)T^{2} \) |
| 23 | \( 1 + (3.91 - 0.425i)T + (22.4 - 4.94i)T^{2} \) |
| 29 | \( 1 + (1.76 - 8.03i)T + (-26.3 - 12.1i)T^{2} \) |
| 31 | \( 1 + (-0.880 - 0.350i)T + (22.5 + 21.3i)T^{2} \) |
| 37 | \( 1 + (3.53 - 4.64i)T + (-9.89 - 35.6i)T^{2} \) |
| 41 | \( 1 + (1.15 - 10.6i)T + (-40.0 - 8.81i)T^{2} \) |
| 43 | \( 1 + (9.57 + 8.13i)T + (6.95 + 42.4i)T^{2} \) |
| 47 | \( 1 + (1.98 + 7.15i)T + (-40.2 + 24.2i)T^{2} \) |
| 53 | \( 1 + (-3.54 + 0.192i)T + (52.6 - 5.73i)T^{2} \) |
| 61 | \( 1 + (-0.571 - 2.59i)T + (-55.3 + 25.6i)T^{2} \) |
| 67 | \( 1 + (3.18 + 4.18i)T + (-17.9 + 64.5i)T^{2} \) |
| 71 | \( 1 + (-14.0 + 3.90i)T + (60.8 - 36.6i)T^{2} \) |
| 73 | \( 1 + (-1.67 - 1.13i)T + (27.0 + 67.8i)T^{2} \) |
| 79 | \( 1 + (1.70 + 10.3i)T + (-74.8 + 25.2i)T^{2} \) |
| 83 | \( 1 + (-2.59 - 4.89i)T + (-46.5 + 68.6i)T^{2} \) |
| 89 | \( 1 + (6.54 + 1.44i)T + (80.7 + 37.3i)T^{2} \) |
| 97 | \( 1 + (-15.0 + 10.2i)T + (35.9 - 90.1i)T^{2} \) |
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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
−12.82782950456954018972089317281, −11.95816770051355978057716453684, −10.23446115390646287678658626534, −9.573183342947517395296366774650, −8.553906103236445475317602664990, −7.53640737643030778156339150260, −6.51414293256163025169539704808, −5.10899726195268390263565627641, −3.35530813088812428314974975353, −1.66619826805815693085016436813,
1.97012246793594672703456103715, 3.98869923487517897269760640436, 5.17063196937995981000457009606, 6.04203202799275318365902364605, 8.163233051052300377883557105776, 9.025752354279742105015815274247, 9.602551424890733775629750321610, 10.36681692614239483394102359242, 11.58725255270496372097115281653, 13.13052428011808180118401130234
