| L(s) = 1 | + (−0.959 − 0.281i)2-s + (−0.755 − 0.654i)3-s + (0.841 + 0.540i)4-s + (−0.288 + 2.00i)5-s + (0.540 + 0.841i)6-s + (2.63 + 0.224i)7-s + (−0.654 − 0.755i)8-s + (0.142 + 0.989i)9-s + (0.840 − 1.84i)10-s + (1.41 + 4.82i)11-s + (−0.281 − 0.959i)12-s + (2.71 + 1.24i)13-s + (−2.46 − 0.958i)14-s + (1.52 − 1.32i)15-s + (0.415 + 0.909i)16-s + (−2.66 + 1.71i)17-s + ⋯ |
| L(s) = 1 | + (−0.678 − 0.199i)2-s + (−0.436 − 0.378i)3-s + (0.420 + 0.270i)4-s + (−0.128 + 0.896i)5-s + (0.220 + 0.343i)6-s + (0.996 + 0.0849i)7-s + (−0.231 − 0.267i)8-s + (0.0474 + 0.329i)9-s + (0.265 − 0.582i)10-s + (0.426 + 1.45i)11-s + (−0.0813 − 0.276i)12-s + (0.753 + 0.344i)13-s + (−0.659 − 0.256i)14-s + (0.395 − 0.342i)15-s + (0.103 + 0.227i)16-s + (−0.646 + 0.415i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0373 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0373 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.629764 + 0.606653i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.629764 + 0.606653i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.755 + 0.654i)T \) |
| 7 | \( 1 + (-2.63 - 0.224i)T \) |
| 23 | \( 1 + (-1.21 + 4.64i)T \) |
| good | 5 | \( 1 + (0.288 - 2.00i)T + (-4.79 - 1.40i)T^{2} \) |
| 11 | \( 1 + (-1.41 - 4.82i)T + (-9.25 + 5.94i)T^{2} \) |
| 13 | \( 1 + (-2.71 - 1.24i)T + (8.51 + 9.82i)T^{2} \) |
| 17 | \( 1 + (2.66 - 1.71i)T + (7.06 - 15.4i)T^{2} \) |
| 19 | \( 1 + (5.50 + 3.54i)T + (7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (7.83 - 5.03i)T + (12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (6.73 - 5.83i)T + (4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (-8.87 + 1.27i)T + (35.5 - 10.4i)T^{2} \) |
| 41 | \( 1 + (-3.87 - 0.557i)T + (39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (6.98 + 6.05i)T + (6.11 + 42.5i)T^{2} \) |
| 47 | \( 1 - 8.39iT - 47T^{2} \) |
| 53 | \( 1 + (3.30 - 1.50i)T + (34.7 - 40.0i)T^{2} \) |
| 59 | \( 1 + (0.0236 + 0.0108i)T + (38.6 + 44.5i)T^{2} \) |
| 61 | \( 1 + (5.64 + 6.51i)T + (-8.68 + 60.3i)T^{2} \) |
| 67 | \( 1 + (0.852 - 2.90i)T + (-56.3 - 36.2i)T^{2} \) |
| 71 | \( 1 + (-7.65 - 2.24i)T + (59.7 + 38.3i)T^{2} \) |
| 73 | \( 1 + (6.38 - 9.92i)T + (-30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (-8.35 - 3.81i)T + (51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (-0.168 - 1.17i)T + (-79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (-0.733 + 0.846i)T + (-12.6 - 88.0i)T^{2} \) |
| 97 | \( 1 + (1.87 - 13.0i)T + (-93.0 - 27.3i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.59543155165353438285564793764, −9.268815798439647476787243831542, −8.650111540213567732244892234528, −7.58116688540558416142058974302, −6.92007235492343515716030080488, −6.34833605603480872461121991203, −4.91412743381627423520685600406, −3.98247465735585397859963864204, −2.38538599168061248638899666573, −1.58538790348301609438106000406,
0.55384128528668984401515911769, 1.76958420869408935147720213602, 3.63851552179641897196509735600, 4.56833238435392747229577786310, 5.69220623002353212148370609761, 6.14273539275817505102733134142, 7.58540221875975100243655843357, 8.314395972037528105809283081051, 8.884965714293737343258779748436, 9.623220375339905453795940096865
