| L(s) = 1 | + (0.285 + 1.38i)2-s + (−2.07 − 1.30i)3-s + (−1.83 + 0.790i)4-s + (−2.72 + 2.17i)5-s + (1.21 − 3.25i)6-s + (0.190 − 0.0435i)7-s + (−1.61 − 2.31i)8-s + (1.31 + 2.73i)9-s + (−3.78 − 3.14i)10-s + (−4.11 + 1.44i)11-s + (4.85 + 0.756i)12-s + (0.215 − 0.447i)13-s + (0.114 + 0.251i)14-s + (8.49 − 0.957i)15-s + (2.74 − 2.90i)16-s + (3.07 + 3.07i)17-s + ⋯ |
| L(s) = 1 | + (0.201 + 0.979i)2-s + (−1.20 − 0.754i)3-s + (−0.918 + 0.395i)4-s + (−1.21 + 0.970i)5-s + (0.496 − 1.32i)6-s + (0.0721 − 0.0164i)7-s + (−0.572 − 0.819i)8-s + (0.438 + 0.910i)9-s + (−1.19 − 0.996i)10-s + (−1.24 + 0.434i)11-s + (1.40 + 0.218i)12-s + (0.0598 − 0.124i)13-s + (0.0306 + 0.0673i)14-s + (2.19 − 0.247i)15-s + (0.687 − 0.726i)16-s + (0.746 + 0.746i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.137i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.990 + 0.137i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0186549 - 0.270510i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0186549 - 0.270510i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.285 - 1.38i)T \) |
| 29 | \( 1 + (4.83 + 2.36i)T \) |
| good | 3 | \( 1 + (2.07 + 1.30i)T + (1.30 + 2.70i)T^{2} \) |
| 5 | \( 1 + (2.72 - 2.17i)T + (1.11 - 4.87i)T^{2} \) |
| 7 | \( 1 + (-0.190 + 0.0435i)T + (6.30 - 3.03i)T^{2} \) |
| 11 | \( 1 + (4.11 - 1.44i)T + (8.60 - 6.85i)T^{2} \) |
| 13 | \( 1 + (-0.215 + 0.447i)T + (-8.10 - 10.1i)T^{2} \) |
| 17 | \( 1 + (-3.07 - 3.07i)T + 17iT^{2} \) |
| 19 | \( 1 + (-3.92 - 6.24i)T + (-8.24 + 17.1i)T^{2} \) |
| 23 | \( 1 + (3.53 + 2.81i)T + (5.11 + 22.4i)T^{2} \) |
| 31 | \( 1 + (5.78 + 0.652i)T + (30.2 + 6.89i)T^{2} \) |
| 37 | \( 1 + (-1.88 + 5.39i)T + (-28.9 - 23.0i)T^{2} \) |
| 41 | \( 1 + (4.93 - 4.93i)T - 41iT^{2} \) |
| 43 | \( 1 + (-0.162 - 1.43i)T + (-41.9 + 9.56i)T^{2} \) |
| 47 | \( 1 + (-1.36 - 3.90i)T + (-36.7 + 29.3i)T^{2} \) |
| 53 | \( 1 + (-0.304 - 0.382i)T + (-11.7 + 51.6i)T^{2} \) |
| 59 | \( 1 + 1.54iT - 59T^{2} \) |
| 61 | \( 1 + (1.69 - 2.70i)T + (-26.4 - 54.9i)T^{2} \) |
| 67 | \( 1 + (-2.69 + 1.29i)T + (41.7 - 52.3i)T^{2} \) |
| 71 | \( 1 + (-10.6 - 5.13i)T + (44.2 + 55.5i)T^{2} \) |
| 73 | \( 1 + (0.531 + 4.72i)T + (-71.1 + 16.2i)T^{2} \) |
| 79 | \( 1 + (2.20 - 6.28i)T + (-61.7 - 49.2i)T^{2} \) |
| 83 | \( 1 + (-10.5 - 2.40i)T + (74.7 + 36.0i)T^{2} \) |
| 89 | \( 1 + (0.947 - 8.41i)T + (-86.7 - 19.8i)T^{2} \) |
| 97 | \( 1 + (-4.31 - 6.86i)T + (-42.0 + 87.3i)T^{2} \) |
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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
−14.31300406026972504472292432386, −12.82660389918785013704560379579, −12.25042857230553214779454587185, −11.17638273319365728215338602650, −10.09472131233762926512386440619, −7.75742623023002325104103081811, −7.69889981590978594892354968068, −6.33883308529944207994994962664, −5.37988023489349958982595022248, −3.69915502209217259685126024775,
0.32001392848983585401070971568, 3.54459065989032122701755837621, 4.94557267452809313638468133111, 5.33714855562526231451919189216, 7.73846603615467486529198738120, 9.038809420174732456116869548845, 10.16793068988167359147397711900, 11.31774517790097296776227739624, 11.63027650000217678681848159237, 12.64867560438175921037738710171
