| L(s) = 1 | + (−0.959 + 0.281i)2-s + (−0.654 − 0.755i)3-s + (0.841 − 0.540i)4-s + (−0.273 − 1.89i)5-s + (0.841 + 0.540i)6-s + (−0.415 + 0.909i)7-s + (−0.654 + 0.755i)8-s + (−0.142 + 0.989i)9-s + (0.797 + 1.74i)10-s + (0.992 + 0.291i)11-s + (−0.959 − 0.281i)12-s + (−0.929 − 2.03i)13-s + (0.142 − 0.989i)14-s + (−1.25 + 1.45i)15-s + (0.415 − 0.909i)16-s + (−2.87 − 1.85i)17-s + ⋯ |
| L(s) = 1 | + (−0.678 + 0.199i)2-s + (−0.378 − 0.436i)3-s + (0.420 − 0.270i)4-s + (−0.122 − 0.849i)5-s + (0.343 + 0.220i)6-s + (−0.157 + 0.343i)7-s + (−0.231 + 0.267i)8-s + (−0.0474 + 0.329i)9-s + (0.252 + 0.551i)10-s + (0.299 + 0.0878i)11-s + (−0.276 − 0.0813i)12-s + (−0.257 − 0.564i)13-s + (0.0380 − 0.264i)14-s + (−0.324 + 0.374i)15-s + (0.103 − 0.227i)16-s + (−0.698 − 0.448i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 + 0.170i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.985 + 0.170i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0314121 - 0.365408i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0314121 - 0.365408i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.959 - 0.281i)T \) |
| 3 | \( 1 + (0.654 + 0.755i)T \) |
| 7 | \( 1 + (0.415 - 0.909i)T \) |
| 23 | \( 1 + (-2.94 + 3.78i)T \) |
| good | 5 | \( 1 + (0.273 + 1.89i)T + (-4.79 + 1.40i)T^{2} \) |
| 11 | \( 1 + (-0.992 - 0.291i)T + (9.25 + 5.94i)T^{2} \) |
| 13 | \( 1 + (0.929 + 2.03i)T + (-8.51 + 9.82i)T^{2} \) |
| 17 | \( 1 + (2.87 + 1.85i)T + (7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (-0.629 + 0.404i)T + (7.89 - 17.2i)T^{2} \) |
| 29 | \( 1 + (7.27 + 4.67i)T + (12.0 + 26.3i)T^{2} \) |
| 31 | \( 1 + (2.57 - 2.97i)T + (-4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (0.937 - 6.51i)T + (-35.5 - 10.4i)T^{2} \) |
| 41 | \( 1 + (0.732 + 5.09i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (-0.387 - 0.447i)T + (-6.11 + 42.5i)T^{2} \) |
| 47 | \( 1 + 12.5T + 47T^{2} \) |
| 53 | \( 1 + (5.36 - 11.7i)T + (-34.7 - 40.0i)T^{2} \) |
| 59 | \( 1 + (1.96 + 4.30i)T + (-38.6 + 44.5i)T^{2} \) |
| 61 | \( 1 + (1.59 - 1.83i)T + (-8.68 - 60.3i)T^{2} \) |
| 67 | \( 1 + (7.94 - 2.33i)T + (56.3 - 36.2i)T^{2} \) |
| 71 | \( 1 + (-10.1 + 2.98i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (10.9 - 7.00i)T + (30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (-1.02 - 2.23i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (-2.02 + 14.0i)T + (-79.6 - 23.3i)T^{2} \) |
| 89 | \( 1 + (7.63 + 8.81i)T + (-12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 + (-0.221 - 1.54i)T + (-93.0 + 27.3i)T^{2} \) |
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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
−9.402597892020577193251097268170, −8.824688450782212166226254905002, −8.020050395786516732650165869128, −7.14509867494702243573375684708, −6.32712756184717743645662633101, −5.35936014468990209495981291711, −4.55260282574126319429585505115, −2.90243154316813162568802150793, −1.57122689376076537224290329335, −0.22234529855807208403131920822,
1.72689836628313104069022282731, 3.16788752574002620008083346738, 3.96217732777291015743875434165, 5.23682049483514218480304824056, 6.44511256857252081875709914175, 6.99653538998020617572130535887, 7.85858602110054882939503851123, 9.074353322754805284908226347861, 9.542282143223888521663296615197, 10.48874845427217768437383470792
