| L(s) = 1 | + (−1.01 + 0.982i)2-s + (−0.984 − 0.173i)3-s + (0.0709 − 1.99i)4-s + (2.00 − 2.38i)5-s + (1.17 − 0.790i)6-s + (−0.650 − 1.12i)7-s + (1.89 + 2.10i)8-s + (0.939 + 0.342i)9-s + (0.306 + 4.40i)10-s + (−3.19 − 1.84i)11-s + (−0.416 + 1.95i)12-s + (2.28 − 0.403i)13-s + (1.76 + 0.508i)14-s + (−2.38 + 2.00i)15-s + (−3.98 − 0.283i)16-s + (−4.34 + 1.58i)17-s + ⋯ |
| L(s) = 1 | + (−0.719 + 0.694i)2-s + (−0.568 − 0.100i)3-s + (0.0354 − 0.999i)4-s + (0.896 − 1.06i)5-s + (0.478 − 0.322i)6-s + (−0.246 − 0.426i)7-s + (0.668 + 0.743i)8-s + (0.313 + 0.114i)9-s + (0.0969 + 1.39i)10-s + (−0.962 − 0.555i)11-s + (−0.120 + 0.564i)12-s + (0.634 − 0.111i)13-s + (0.472 + 0.135i)14-s + (−0.616 + 0.517i)15-s + (−0.997 − 0.0709i)16-s + (−1.05 + 0.383i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.130 + 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.130 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.411451 - 0.469246i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.411451 - 0.469246i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.01 - 0.982i)T \) |
| 3 | \( 1 + (0.984 + 0.173i)T \) |
| 19 | \( 1 + (1.36 + 4.13i)T \) |
| good | 5 | \( 1 + (-2.00 + 2.38i)T + (-0.868 - 4.92i)T^{2} \) |
| 7 | \( 1 + (0.650 + 1.12i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (3.19 + 1.84i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.28 + 0.403i)T + (12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (4.34 - 1.58i)T + (13.0 - 10.9i)T^{2} \) |
| 23 | \( 1 + (4.32 - 3.62i)T + (3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (-0.617 + 1.69i)T + (-22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (2.13 + 3.70i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 0.384iT - 37T^{2} \) |
| 41 | \( 1 + (0.439 - 2.49i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (-7.88 + 9.39i)T + (-7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (-1.33 - 0.484i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + (2.65 + 3.16i)T + (-9.20 + 52.1i)T^{2} \) |
| 59 | \( 1 + (-0.866 - 2.38i)T + (-45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (5.61 + 6.69i)T + (-10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (-0.167 + 0.461i)T + (-51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (-0.995 - 0.835i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (0.785 - 4.45i)T + (-68.5 - 24.9i)T^{2} \) |
| 79 | \( 1 + (1.91 - 10.8i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (11.6 - 6.74i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (3.11 + 17.6i)T + (-83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (-17.8 + 6.48i)T + (74.3 - 62.3i)T^{2} \) |
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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
−10.66307103083448328142898388309, −9.829221695029329569784223584743, −8.958464755133981674637587018248, −8.231446114798965861494230829915, −7.09659501654591276630090503475, −6.00806479440219198514366746108, −5.49902444431008411567191518342, −4.38729442877991963908715985445, −2.00036143924140594145786624222, −0.50333316298690369728668287152,
1.97297588803887835281555207299, 2.90228339292153165762050161731, 4.38870565380059785532278473112, 5.90268168865727456016229434378, 6.66256347068732511671378771615, 7.69825975736459989673893032911, 8.887187943654801477874253474068, 9.783111181696352665404222442425, 10.58466005289532674139334696378, 10.85845653828985182636404595952
