| 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
−10.85845653828985182636404595952, −10.58466005289532674139334696378, −9.783111181696352665404222442425, −8.887187943654801477874253474068, −7.69825975736459989673893032911, −6.66256347068732511671378771615, −5.90268168865727456016229434378, −4.38870565380059785532278473112, −2.90228339292153165762050161731, −1.97297588803887835281555207299,
0.50333316298690369728668287152, 2.00036143924140594145786624222, 4.38729442877991963908715985445, 5.49902444431008411567191518342, 6.00806479440219198514366746108, 7.09659501654591276630090503475, 8.231446114798965861494230829915, 8.958464755133981674637587018248, 9.829221695029329569784223584743, 10.66307103083448328142898388309
