| L(s) = 1 | + (1.32 + 0.506i)2-s + (−1.02 − 1.39i)3-s + (1.48 + 1.33i)4-s + (−0.779 + 2.14i)5-s + (−0.643 − 2.36i)6-s + (−0.984 − 0.173i)7-s + (1.28 + 2.51i)8-s + (−0.904 + 2.86i)9-s + (−2.11 + 2.43i)10-s + (−5.69 + 2.07i)11-s + (0.347 − 3.44i)12-s + (2.72 − 2.28i)13-s + (−1.21 − 0.728i)14-s + (3.78 − 1.10i)15-s + (0.419 + 3.97i)16-s + (−5.84 + 3.37i)17-s + ⋯ |
| L(s) = 1 | + (0.933 + 0.358i)2-s + (−0.590 − 0.806i)3-s + (0.743 + 0.668i)4-s + (−0.348 + 0.957i)5-s + (−0.262 − 0.964i)6-s + (−0.372 − 0.0656i)7-s + (0.454 + 0.890i)8-s + (−0.301 + 0.953i)9-s + (−0.668 + 0.769i)10-s + (−1.71 + 0.624i)11-s + (0.100 − 0.994i)12-s + (0.755 − 0.634i)13-s + (−0.324 − 0.194i)14-s + (0.978 − 0.284i)15-s + (0.104 + 0.994i)16-s + (−1.41 + 0.818i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.686 - 0.727i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.686 - 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.474281 + 1.09992i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.474281 + 1.09992i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.32 - 0.506i)T \) |
| 3 | \( 1 + (1.02 + 1.39i)T \) |
| 7 | \( 1 + (0.984 + 0.173i)T \) |
| good | 5 | \( 1 + (0.779 - 2.14i)T + (-3.83 - 3.21i)T^{2} \) |
| 11 | \( 1 + (5.69 - 2.07i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (-2.72 + 2.28i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (5.84 - 3.37i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.21 + 0.701i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.430 + 2.44i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (1.90 - 2.26i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (1.27 - 0.225i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-4.61 - 7.99i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (1.27 + 1.52i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-2.90 - 7.99i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-1.14 + 6.50i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 - 5.43iT - 53T^{2} \) |
| 59 | \( 1 + (-12.9 - 4.70i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-1.52 + 8.64i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (3.52 + 4.19i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (1.92 + 3.34i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (1.77 - 3.07i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (0.893 - 1.06i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-6.03 - 5.06i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (11.5 + 6.66i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-7.40 + 2.69i)T + (74.3 - 62.3i)T^{2} \) |
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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
−10.77957678853338355156334779208, −10.42720242575493014997606361631, −8.439523397076170981973363407139, −7.77255330145582925587915140455, −6.94240900300040661406409503460, −6.36373573386643312589314993719, −5.45943951187640879269419542032, −4.43629850501772551288492308979, −3.06533527883076077416118015834, −2.22650668996051079821141140165,
0.44505356753102030340980809390, 2.49542678149644397591381648106, 3.78640638507148660235362534919, 4.53674485612440042809163477242, 5.37583204347369767362330718539, 6.03027461095541863164458008672, 7.18564840662617831780769791909, 8.535241188113525312524870494601, 9.310368010827952984950128878377, 10.29585571085408783214996723370
