| L(s) = 1 | + (−0.959 + 0.281i)2-s + (−0.923 − 1.06i)3-s + (0.841 − 0.540i)4-s + (−0.555 − 3.86i)5-s + (1.18 + 0.762i)6-s + (−0.415 + 0.909i)7-s + (−0.654 + 0.755i)8-s + (0.143 − 0.999i)9-s + (1.62 + 3.55i)10-s + (−3.85 − 1.13i)11-s + (−1.35 − 0.397i)12-s + (1.61 + 3.54i)13-s + (0.142 − 0.989i)14-s + (−3.60 + 4.16i)15-s + (0.415 − 0.909i)16-s + (2.53 + 1.63i)17-s + ⋯ |
| L(s) = 1 | + (−0.678 + 0.199i)2-s + (−0.533 − 0.615i)3-s + (0.420 − 0.270i)4-s + (−0.248 − 1.72i)5-s + (0.484 + 0.311i)6-s + (−0.157 + 0.343i)7-s + (−0.231 + 0.267i)8-s + (0.0478 − 0.333i)9-s + (0.513 + 1.12i)10-s + (−1.16 − 0.341i)11-s + (−0.390 − 0.114i)12-s + (0.448 + 0.982i)13-s + (0.0380 − 0.264i)14-s + (−0.931 + 1.07i)15-s + (0.103 − 0.227i)16-s + (0.615 + 0.395i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.105i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 + 0.105i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0214101 - 0.405767i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0214101 - 0.405767i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.959 - 0.281i)T \) |
| 7 | \( 1 + (0.415 - 0.909i)T \) |
| 23 | \( 1 + (0.614 + 4.75i)T \) |
| good | 3 | \( 1 + (0.923 + 1.06i)T + (-0.426 + 2.96i)T^{2} \) |
| 5 | \( 1 + (0.555 + 3.86i)T + (-4.79 + 1.40i)T^{2} \) |
| 11 | \( 1 + (3.85 + 1.13i)T + (9.25 + 5.94i)T^{2} \) |
| 13 | \( 1 + (-1.61 - 3.54i)T + (-8.51 + 9.82i)T^{2} \) |
| 17 | \( 1 + (-2.53 - 1.63i)T + (7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (4.43 - 2.84i)T + (7.89 - 17.2i)T^{2} \) |
| 29 | \( 1 + (2.94 + 1.89i)T + (12.0 + 26.3i)T^{2} \) |
| 31 | \( 1 + (-0.565 + 0.652i)T + (-4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (-1.11 + 7.73i)T + (-35.5 - 10.4i)T^{2} \) |
| 41 | \( 1 + (-0.0744 - 0.517i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (5.33 + 6.15i)T + (-6.11 + 42.5i)T^{2} \) |
| 47 | \( 1 - 5.22T + 47T^{2} \) |
| 53 | \( 1 + (0.565 - 1.23i)T + (-34.7 - 40.0i)T^{2} \) |
| 59 | \( 1 + (-3.54 - 7.75i)T + (-38.6 + 44.5i)T^{2} \) |
| 61 | \( 1 + (0.872 - 1.00i)T + (-8.68 - 60.3i)T^{2} \) |
| 67 | \( 1 + (-4.00 + 1.17i)T + (56.3 - 36.2i)T^{2} \) |
| 71 | \( 1 + (11.9 - 3.49i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (-11.4 + 7.36i)T + (30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (-1.39 - 3.06i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (-2.10 + 14.6i)T + (-79.6 - 23.3i)T^{2} \) |
| 89 | \( 1 + (5.33 + 6.15i)T + (-12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 + (-0.522 - 3.63i)T + (-93.0 + 27.3i)T^{2} \) |
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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
−11.29220475450304373002880508005, −10.14053872121343044576805937257, −8.983218958470621492890727471067, −8.482246615997271898231697126165, −7.52074781971501437854696777103, −6.17514971101000285111451889086, −5.49913796332303578705922659550, −4.11847669098331178610887890631, −1.82027166021176154261405785030, −0.37160470244822533461342070302,
2.55639123108544580935401771575, 3.57508412091696323641108592166, 5.18625291204795782972893660386, 6.40487183396571212962804507350, 7.45173498327788215993226357930, 8.060582862542847672573151600451, 9.822694835219300977351873530540, 10.31554240298651706314914012911, 10.93129667897128629363769459663, 11.46232055995209850862760092512
