L(s) = 1 | + (1.20 + 0.743i)2-s + (0.818 + 1.41i)3-s + (0.894 + 1.78i)4-s + (2.22 − 0.187i)5-s + (−0.0691 + 2.31i)6-s + (−2.64 − 0.148i)7-s + (−0.253 + 2.81i)8-s + (0.158 − 0.274i)9-s + (2.82 + 1.43i)10-s + (−2.19 − 3.79i)11-s + (−1.80 + 2.73i)12-s − 2.75i·13-s + (−3.06 − 2.14i)14-s + (2.09 + 3.00i)15-s + (−2.39 + 3.20i)16-s + (0.648 + 1.12i)17-s + ⋯ |
L(s) = 1 | + (0.850 + 0.525i)2-s + (0.472 + 0.818i)3-s + (0.447 + 0.894i)4-s + (0.996 − 0.0840i)5-s + (−0.0282 + 0.945i)6-s + (−0.998 − 0.0561i)7-s + (−0.0895 + 0.995i)8-s + (0.0529 − 0.0916i)9-s + (0.891 + 0.452i)10-s + (−0.661 − 1.14i)11-s + (−0.520 + 0.789i)12-s − 0.764i·13-s + (−0.819 − 0.572i)14-s + (0.539 + 0.776i)15-s + (−0.599 + 0.800i)16-s + (0.157 + 0.272i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.187 - 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.187 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.83223 + 1.51592i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.83223 + 1.51592i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.20 - 0.743i)T \) |
| 5 | \( 1 + (-2.22 + 0.187i)T \) |
| 7 | \( 1 + (2.64 + 0.148i)T \) |
good | 3 | \( 1 + (-0.818 - 1.41i)T + (-1.5 + 2.59i)T^{2} \) |
| 11 | \( 1 + (2.19 + 3.79i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 2.75iT - 13T^{2} \) |
| 17 | \( 1 + (-0.648 - 1.12i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (3.86 + 2.23i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.136 - 0.236i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 9.84iT - 29T^{2} \) |
| 31 | \( 1 + (2.27 + 3.94i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (3.94 - 6.83i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 3.95iT - 41T^{2} \) |
| 43 | \( 1 - 4.46iT - 43T^{2} \) |
| 47 | \( 1 + (-7.22 - 4.17i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (5.40 + 9.35i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.31 + 1.91i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.73 + 8.20i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (8.32 - 4.80i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 1.42iT - 71T^{2} \) |
| 73 | \( 1 + (1.84 + 3.19i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-12.3 - 7.11i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 12.8T + 83T^{2} \) |
| 89 | \( 1 + (3.15 + 1.82i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 0.0199T + 97T^{2} \) |
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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
−12.61483782018886719164230987463, −10.92724550496890567371646561838, −10.22743660824293303334222928891, −9.116711151125332350173090474968, −8.362620117267856834524759869802, −6.82924128958343266871342801532, −5.95589418271471653110331250406, −5.03087899302329513625755038607, −3.57193407673272565650061676464, −2.81574226457793270901893377038,
1.90540485731660223332647984743, 2.62251646816365438621156894088, 4.29055313412160273455203478978, 5.63828290775879218812443082328, 6.62680868678907448259024417043, 7.37182947087181693080644194985, 9.047681157208689775642267890522, 9.990282230791470283580077082187, 10.57382237392615660149471319485, 12.15426440751422609246832025205