| L(s) = 1 | + (−0.873 + 2.60i)2-s + (−2.35 − 1.32i)3-s + (−4.45 − 3.35i)4-s + (0.433 − 0.654i)5-s + (5.51 − 4.99i)6-s + (−1.45 + 3.21i)7-s + (8.11 − 5.56i)8-s + (2.24 + 3.68i)9-s + (1.32 + 1.70i)10-s + (−2.67 − 0.171i)11-s + (6.04 + 13.8i)12-s + (3.71 + 4.29i)13-s + (−7.10 − 6.60i)14-s + (−1.88 + 0.967i)15-s + (4.37 + 15.3i)16-s + (−2.36 + 1.04i)17-s + ⋯ |
| L(s) = 1 | + (−0.617 + 1.84i)2-s + (−1.36 − 0.764i)3-s + (−2.22 − 1.67i)4-s + (0.193 − 0.292i)5-s + (2.25 − 2.03i)6-s + (−0.549 + 1.21i)7-s + (2.86 − 1.96i)8-s + (0.747 + 1.22i)9-s + (0.420 + 0.538i)10-s + (−0.806 − 0.0516i)11-s + (1.74 + 3.98i)12-s + (1.03 + 1.19i)13-s + (−1.89 − 1.76i)14-s + (−0.487 + 0.249i)15-s + (1.09 + 3.82i)16-s + (−0.572 + 0.252i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 983 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.899 + 0.436i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 983 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.899 + 0.436i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.109853 - 0.0252269i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.109853 - 0.0252269i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 983 | \( 1 + (15.0 + 27.5i)T \) |
| good | 2 | \( 1 + (0.873 - 2.60i)T + (-1.59 - 1.20i)T^{2} \) |
| 3 | \( 1 + (2.35 + 1.32i)T + (1.56 + 2.56i)T^{2} \) |
| 5 | \( 1 + (-0.433 + 0.654i)T + (-1.94 - 4.60i)T^{2} \) |
| 7 | \( 1 + (1.45 - 3.21i)T + (-4.61 - 5.26i)T^{2} \) |
| 11 | \( 1 + (2.67 + 0.171i)T + (10.9 + 1.40i)T^{2} \) |
| 13 | \( 1 + (-3.71 - 4.29i)T + (-1.86 + 12.8i)T^{2} \) |
| 17 | \( 1 + (2.36 - 1.04i)T + (11.4 - 12.5i)T^{2} \) |
| 19 | \( 1 + (5.97 + 4.56i)T + (4.98 + 18.3i)T^{2} \) |
| 23 | \( 1 + (-0.529 - 4.99i)T + (-22.4 + 4.82i)T^{2} \) |
| 29 | \( 1 + (1.27 - 1.85i)T + (-10.2 - 27.1i)T^{2} \) |
| 31 | \( 1 + (4.36 + 6.49i)T + (-11.7 + 28.7i)T^{2} \) |
| 37 | \( 1 + (0.993 - 0.246i)T + (32.7 - 17.2i)T^{2} \) |
| 41 | \( 1 + (-6.68 - 0.600i)T + (40.3 + 7.30i)T^{2} \) |
| 43 | \( 1 + (-3.20 - 4.39i)T + (-13.1 + 40.9i)T^{2} \) |
| 47 | \( 1 + (1.38 + 4.63i)T + (-39.2 + 25.8i)T^{2} \) |
| 53 | \( 1 + (-3.74 + 3.43i)T + (4.57 - 52.8i)T^{2} \) |
| 59 | \( 1 + (7.36 + 0.566i)T + (58.3 + 9.02i)T^{2} \) |
| 61 | \( 1 + (-1.51 - 1.28i)T + (9.90 + 60.1i)T^{2} \) |
| 67 | \( 1 + (-3.66 + 3.14i)T + (10.0 - 66.2i)T^{2} \) |
| 71 | \( 1 + (-5.78 + 4.72i)T + (14.2 - 69.5i)T^{2} \) |
| 73 | \( 1 + (8.05 + 5.75i)T + (23.6 + 69.0i)T^{2} \) |
| 79 | \( 1 + (-11.4 + 1.70i)T + (75.6 - 22.9i)T^{2} \) |
| 83 | \( 1 + (0.599 - 0.185i)T + (68.4 - 46.9i)T^{2} \) |
| 89 | \( 1 + (13.5 + 8.91i)T + (35.1 + 81.7i)T^{2} \) |
| 97 | \( 1 + (16.1 - 0.825i)T + (96.4 - 9.91i)T^{2} \) |
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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
−9.334350340791773886236913783197, −9.078445885284040663634993395349, −8.130270862542397674792378174340, −7.07304006467810172818548833761, −6.52212080900411169483616375952, −5.83496483519858028167070867213, −5.40557998147533029285582501759, −4.37547388851425834491252206722, −1.73840395120620984967134655825, −0.10969128430113070821074534646,
0.873953188716713616579699897669, 2.61174448361302959284969398750, 3.81022270391135282700322921379, 4.31278189544388326819161271442, 5.42819894095684554582425041198, 6.55905098248787429358570358304, 7.88805374951625603948883559027, 8.749382582537556729804277118451, 9.883278051731653247362529115666, 10.49944474267532639696183217807
