L(s) = 1 | + (−0.654 − 0.755i)2-s + (−0.841 − 0.540i)3-s + (−0.142 + 0.989i)4-s + (0.479 − 1.05i)5-s + (0.142 + 0.989i)6-s + (0.959 + 0.281i)7-s + (0.841 − 0.540i)8-s + (0.415 + 0.909i)9-s + (−1.10 + 0.325i)10-s + (1.74 − 2.01i)11-s + (0.654 − 0.755i)12-s + (2.97 − 0.872i)13-s + (−0.415 − 0.909i)14-s + (−0.971 + 0.624i)15-s + (−0.959 − 0.281i)16-s + (0.910 + 6.33i)17-s + ⋯ |
L(s) = 1 | + (−0.463 − 0.534i)2-s + (−0.485 − 0.312i)3-s + (−0.0711 + 0.494i)4-s + (0.214 − 0.469i)5-s + (0.0580 + 0.404i)6-s + (0.362 + 0.106i)7-s + (0.297 − 0.191i)8-s + (0.138 + 0.303i)9-s + (−0.350 + 0.102i)10-s + (0.526 − 0.607i)11-s + (0.189 − 0.218i)12-s + (0.824 − 0.241i)13-s + (−0.111 − 0.243i)14-s + (−0.250 + 0.161i)15-s + (−0.239 − 0.0704i)16-s + (0.220 + 1.53i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.486 + 0.873i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.486 + 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.05233 - 0.618350i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.05233 - 0.618350i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.654 + 0.755i)T \) |
| 3 | \( 1 + (0.841 + 0.540i)T \) |
| 7 | \( 1 + (-0.959 - 0.281i)T \) |
| 23 | \( 1 + (1.66 + 4.49i)T \) |
good | 5 | \( 1 + (-0.479 + 1.05i)T + (-3.27 - 3.77i)T^{2} \) |
| 11 | \( 1 + (-1.74 + 2.01i)T + (-1.56 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-2.97 + 0.872i)T + (10.9 - 7.02i)T^{2} \) |
| 17 | \( 1 + (-0.910 - 6.33i)T + (-16.3 + 4.78i)T^{2} \) |
| 19 | \( 1 + (0.717 - 4.98i)T + (-18.2 - 5.35i)T^{2} \) |
| 29 | \( 1 + (0.136 + 0.947i)T + (-27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (-3.04 + 1.95i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + (-2.81 - 6.17i)T + (-24.2 + 27.9i)T^{2} \) |
| 41 | \( 1 + (-4.74 + 10.3i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (1.82 + 1.17i)T + (17.8 + 39.1i)T^{2} \) |
| 47 | \( 1 - 9.44T + 47T^{2} \) |
| 53 | \( 1 + (2.51 + 0.739i)T + (44.5 + 28.6i)T^{2} \) |
| 59 | \( 1 + (-9.31 + 2.73i)T + (49.6 - 31.8i)T^{2} \) |
| 61 | \( 1 + (6.86 - 4.41i)T + (25.3 - 55.4i)T^{2} \) |
| 67 | \( 1 + (2.87 + 3.32i)T + (-9.53 + 66.3i)T^{2} \) |
| 71 | \( 1 + (-0.375 - 0.433i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (-2.04 + 14.2i)T + (-70.0 - 20.5i)T^{2} \) |
| 79 | \( 1 + (2.63 - 0.775i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (5.09 + 11.1i)T + (-54.3 + 62.7i)T^{2} \) |
| 89 | \( 1 + (8.82 + 5.66i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (-4.53 + 9.92i)T + (-63.5 - 73.3i)T^{2} \) |
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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
−10.13066792886756316005726696093, −8.826917056821312138094509595916, −8.462750937688799776771662781125, −7.58549293088129194568035577824, −6.23434958645136688014194701367, −5.79406873984118479895704043674, −4.42343173422435643608736929119, −3.51604187210076428754743836479, −1.92849842611409153645316836341, −0.982168514631209303495676073987,
1.05804912691508501254533678980, 2.66098483121268295443049662508, 4.17179652203186151734014704039, 5.00893771535404647666140260114, 6.00093636704046471566680049564, 6.85099339492623983750058872286, 7.41811796634949082950952775589, 8.594823247961077109500936937407, 9.423539034030116852446291280870, 9.953139470960101978715579374232