| L(s) = 1 | + (0.281 − 0.959i)2-s + (−1.78 − 1.54i)3-s + (−0.841 − 0.540i)4-s + (−0.284 + 2.21i)5-s + (−1.98 + 1.27i)6-s + (−3.91 + 1.78i)7-s + (−0.755 + 0.654i)8-s + (0.363 + 2.52i)9-s + (2.04 + 0.897i)10-s + (−1.25 + 0.368i)11-s + (0.663 + 2.26i)12-s + (3.30 + 1.50i)13-s + (0.612 + 4.26i)14-s + (3.92 − 3.51i)15-s + (0.415 + 0.909i)16-s + (−2.01 − 3.12i)17-s + ⋯ |
| L(s) = 1 | + (0.199 − 0.678i)2-s + (−1.02 − 0.890i)3-s + (−0.420 − 0.270i)4-s + (−0.127 + 0.991i)5-s + (−0.809 + 0.520i)6-s + (−1.48 + 0.675i)7-s + (−0.267 + 0.231i)8-s + (0.121 + 0.841i)9-s + (0.647 + 0.283i)10-s + (−0.378 + 0.111i)11-s + (0.191 + 0.652i)12-s + (0.916 + 0.418i)13-s + (0.163 + 1.13i)14-s + (1.01 − 0.906i)15-s + (0.103 + 0.227i)16-s + (−0.487 − 0.758i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.242 - 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.242 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0647029 + 0.0828994i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0647029 + 0.0828994i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.281 + 0.959i)T \) |
| 5 | \( 1 + (0.284 - 2.21i)T \) |
| 23 | \( 1 + (1.70 - 4.48i)T \) |
| good | 3 | \( 1 + (1.78 + 1.54i)T + (0.426 + 2.96i)T^{2} \) |
| 7 | \( 1 + (3.91 - 1.78i)T + (4.58 - 5.29i)T^{2} \) |
| 11 | \( 1 + (1.25 - 0.368i)T + (9.25 - 5.94i)T^{2} \) |
| 13 | \( 1 + (-3.30 - 1.50i)T + (8.51 + 9.82i)T^{2} \) |
| 17 | \( 1 + (2.01 + 3.12i)T + (-7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (6.15 + 3.95i)T + (7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (1.95 - 1.25i)T + (12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (-3.09 - 3.56i)T + (-4.41 + 30.6i)T^{2} \) |
| 37 | \( 1 + (5.75 - 0.827i)T + (35.5 - 10.4i)T^{2} \) |
| 41 | \( 1 + (-0.430 + 2.99i)T + (-39.3 - 11.5i)T^{2} \) |
| 43 | \( 1 + (2.51 + 2.17i)T + (6.11 + 42.5i)T^{2} \) |
| 47 | \( 1 + 5.43iT - 47T^{2} \) |
| 53 | \( 1 + (-4.88 + 2.23i)T + (34.7 - 40.0i)T^{2} \) |
| 59 | \( 1 + (-2.87 + 6.29i)T + (-38.6 - 44.5i)T^{2} \) |
| 61 | \( 1 + (6.18 + 7.13i)T + (-8.68 + 60.3i)T^{2} \) |
| 67 | \( 1 + (0.940 - 3.20i)T + (-56.3 - 36.2i)T^{2} \) |
| 71 | \( 1 + (-12.8 - 3.78i)T + (59.7 + 38.3i)T^{2} \) |
| 73 | \( 1 + (7.36 - 11.4i)T + (-30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (6.84 - 14.9i)T + (-51.7 - 59.7i)T^{2} \) |
| 83 | \( 1 + (-9.90 + 1.42i)T + (79.6 - 23.3i)T^{2} \) |
| 89 | \( 1 + (6.13 - 7.08i)T + (-12.6 - 88.0i)T^{2} \) |
| 97 | \( 1 + (-4.82 - 0.693i)T + (93.0 + 27.3i)T^{2} \) |
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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
−12.40419600347030287423609064746, −11.54366112210997360805843401746, −10.90103866944135428143512426889, −9.865223540009751848238908272740, −8.734639137232127803078397567335, −6.91481622740968671421590601528, −6.51741698839339171998887594327, −5.47015888492460930137459083436, −3.59683228685767170310876410347, −2.30854817213238955392144240168,
0.086568937536545741996143920999, 3.80623598136747716653243334357, 4.49479097937511160678841403142, 5.91971376869604972101831566909, 6.30593203183303577565043958919, 8.001611880204541890195211118071, 8.997420045198256619460990039841, 10.17558358567636521463835201961, 10.66580285888602594746081767821, 12.12279804843854713084124972836
