| L(s) = 1 | + (−0.668 + 1.99i)2-s + (2.16 + 1.21i)3-s + (−1.93 − 1.46i)4-s + (−1.46 + 2.21i)5-s + (−3.87 + 3.50i)6-s + (−1.51 + 3.34i)7-s + (0.745 − 0.511i)8-s + (1.64 + 2.69i)9-s + (−3.43 − 4.40i)10-s + (0.635 + 0.0407i)11-s + (−2.41 − 5.52i)12-s + (3.77 + 4.36i)13-s + (−5.66 − 5.26i)14-s + (−5.85 + 3.00i)15-s + (−0.812 − 2.84i)16-s + (4.47 − 1.97i)17-s + ⋯ |
| L(s) = 1 | + (−0.472 + 1.41i)2-s + (1.24 + 0.701i)3-s + (−0.969 − 0.731i)4-s + (−0.655 + 0.988i)5-s + (−1.58 + 1.43i)6-s + (−0.573 + 1.26i)7-s + (0.263 − 0.180i)8-s + (0.547 + 0.898i)9-s + (−1.08 − 1.39i)10-s + (0.191 + 0.0122i)11-s + (−0.697 − 1.59i)12-s + (1.04 + 1.21i)13-s + (−1.51 − 1.40i)14-s + (−1.51 + 0.775i)15-s + (−0.203 − 0.710i)16-s + (1.08 − 0.479i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 983 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 983 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.792446 - 1.28187i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.792446 - 1.28187i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 983 | \( 1 + (-27.6 + 14.7i)T \) |
| good | 2 | \( 1 + (0.668 - 1.99i)T + (-1.59 - 1.20i)T^{2} \) |
| 3 | \( 1 + (-2.16 - 1.21i)T + (1.56 + 2.56i)T^{2} \) |
| 5 | \( 1 + (1.46 - 2.21i)T + (-1.94 - 4.60i)T^{2} \) |
| 7 | \( 1 + (1.51 - 3.34i)T + (-4.61 - 5.26i)T^{2} \) |
| 11 | \( 1 + (-0.635 - 0.0407i)T + (10.9 + 1.40i)T^{2} \) |
| 13 | \( 1 + (-3.77 - 4.36i)T + (-1.86 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-4.47 + 1.97i)T + (11.4 - 12.5i)T^{2} \) |
| 19 | \( 1 + (1.93 + 1.47i)T + (4.98 + 18.3i)T^{2} \) |
| 23 | \( 1 + (-0.282 - 2.66i)T + (-22.4 + 4.82i)T^{2} \) |
| 29 | \( 1 + (-3.20 + 4.64i)T + (-10.2 - 27.1i)T^{2} \) |
| 31 | \( 1 + (1.20 + 1.79i)T + (-11.7 + 28.7i)T^{2} \) |
| 37 | \( 1 + (-9.95 + 2.46i)T + (32.7 - 17.2i)T^{2} \) |
| 41 | \( 1 + (4.15 + 0.373i)T + (40.3 + 7.30i)T^{2} \) |
| 43 | \( 1 + (-2.76 - 3.78i)T + (-13.1 + 40.9i)T^{2} \) |
| 47 | \( 1 + (0.834 + 2.78i)T + (-39.2 + 25.8i)T^{2} \) |
| 53 | \( 1 + (-5.11 + 4.69i)T + (4.57 - 52.8i)T^{2} \) |
| 59 | \( 1 + (4.82 + 0.371i)T + (58.3 + 9.02i)T^{2} \) |
| 61 | \( 1 + (2.42 + 2.05i)T + (9.90 + 60.1i)T^{2} \) |
| 67 | \( 1 + (10.3 - 8.90i)T + (10.0 - 66.2i)T^{2} \) |
| 71 | \( 1 + (1.12 - 0.916i)T + (14.2 - 69.5i)T^{2} \) |
| 73 | \( 1 + (-1.77 - 1.26i)T + (23.6 + 69.0i)T^{2} \) |
| 79 | \( 1 + (-3.35 + 0.497i)T + (75.6 - 22.9i)T^{2} \) |
| 83 | \( 1 + (13.3 - 4.13i)T + (68.4 - 46.9i)T^{2} \) |
| 89 | \( 1 + (5.67 + 3.73i)T + (35.1 + 81.7i)T^{2} \) |
| 97 | \( 1 + (-12.2 + 0.626i)T + (96.4 - 9.91i)T^{2} \) |
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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
−9.984218484338188480248111668411, −9.336695147010912316183773449472, −8.830573367122687014732954484330, −8.120014869059170679729655227694, −7.34482892801035550176652734276, −6.44533795625544352009176380648, −5.77890940142293932743204546705, −4.33775956364091208845391036601, −3.32544896044323834150885247861, −2.54169799216856918629411490812,
0.78116823763261328948720910733, 1.40427545397070962777912883249, 3.00040605827942087163500076691, 3.53740405462163876931985627321, 4.36471873122608540310145256149, 6.12755963787901595843348874565, 7.36572831472029722428117478781, 8.201935280203687914321263477156, 8.536672771156150241331776599969, 9.426505398610510006968227024214
