| L(s) = 1 | + (−0.841 + 0.540i)2-s + (−0.142 + 0.989i)3-s + (0.415 − 0.909i)4-s + (−0.421 + 0.123i)5-s + (−0.415 − 0.909i)6-s + (0.654 + 0.755i)7-s + (0.142 + 0.989i)8-s + (−0.959 − 0.281i)9-s + (0.287 − 0.331i)10-s + (−2.89 − 1.86i)11-s + (0.841 + 0.540i)12-s + (1.66 − 1.92i)13-s + (−0.959 − 0.281i)14-s + (−0.0625 − 0.434i)15-s + (−0.654 − 0.755i)16-s + (−2.50 − 5.47i)17-s + ⋯ |
| L(s) = 1 | + (−0.594 + 0.382i)2-s + (−0.0821 + 0.571i)3-s + (0.207 − 0.454i)4-s + (−0.188 + 0.0553i)5-s + (−0.169 − 0.371i)6-s + (0.247 + 0.285i)7-s + (0.0503 + 0.349i)8-s + (−0.319 − 0.0939i)9-s + (0.0909 − 0.104i)10-s + (−0.874 − 0.561i)11-s + (0.242 + 0.156i)12-s + (0.461 − 0.532i)13-s + (−0.256 − 0.0752i)14-s + (−0.0161 − 0.112i)15-s + (−0.163 − 0.188i)16-s + (−0.606 − 1.32i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.211 + 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.211 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.373826 - 0.301450i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.373826 - 0.301450i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.841 - 0.540i)T \) |
| 3 | \( 1 + (0.142 - 0.989i)T \) |
| 7 | \( 1 + (-0.654 - 0.755i)T \) |
| 23 | \( 1 + (-1.49 + 4.55i)T \) |
| good | 5 | \( 1 + (0.421 - 0.123i)T + (4.20 - 2.70i)T^{2} \) |
| 11 | \( 1 + (2.89 + 1.86i)T + (4.56 + 10.0i)T^{2} \) |
| 13 | \( 1 + (-1.66 + 1.92i)T + (-1.85 - 12.8i)T^{2} \) |
| 17 | \( 1 + (2.50 + 5.47i)T + (-11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (2.55 - 5.59i)T + (-12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (2.23 + 4.89i)T + (-18.9 + 21.9i)T^{2} \) |
| 31 | \( 1 + (0.135 + 0.939i)T + (-29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (0.0508 + 0.0149i)T + (31.1 + 20.0i)T^{2} \) |
| 41 | \( 1 + (0.720 - 0.211i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (-0.685 + 4.76i)T + (-41.2 - 12.1i)T^{2} \) |
| 47 | \( 1 + 1.19T + 47T^{2} \) |
| 53 | \( 1 + (1.84 + 2.13i)T + (-7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (-0.680 + 0.784i)T + (-8.39 - 58.3i)T^{2} \) |
| 61 | \( 1 + (1.95 + 13.5i)T + (-58.5 + 17.1i)T^{2} \) |
| 67 | \( 1 + (7.79 - 5.00i)T + (27.8 - 60.9i)T^{2} \) |
| 71 | \( 1 + (-3.15 + 2.02i)T + (29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (-0.616 + 1.35i)T + (-47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (-8.34 + 9.63i)T + (-11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (0.0530 + 0.0155i)T + (69.8 + 44.8i)T^{2} \) |
| 89 | \( 1 + (-0.319 + 2.21i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (-14.9 + 4.40i)T + (81.6 - 52.4i)T^{2} \) |
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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
−9.817080792000338672264509138698, −8.950326010329508352811559056671, −8.195861772037131051561112502826, −7.58275497244344934457610929892, −6.32768524077593191313445478297, −5.59401297010592491458182037866, −4.71074949907331402641133432754, −3.45288435802134462441701355734, −2.23367135791533064318863874138, −0.27014266536833983822481480891,
1.47520013826433046897510059288, 2.46831427280792326610752152346, 3.83427566462829045825839917074, 4.85392324377322647787254188732, 6.13534736913871811779008492237, 7.02610426535421230365658366135, 7.74487914237823535974048209824, 8.541194663925411638046459671823, 9.253824428271924418171853388143, 10.36904042288294549664161876532
