| L(s) = 1 | + (−0.596 − 1.15i)2-s + (−1.67 + 0.439i)3-s + (0.177 − 0.249i)4-s + (0.518 + 2.13i)5-s + (1.50 + 1.67i)6-s + (−2.98 + 1.03i)7-s + (−2.97 − 0.427i)8-s + (2.61 − 1.47i)9-s + (2.16 − 1.87i)10-s + (0.308 + 6.48i)11-s + (−0.188 + 0.496i)12-s + (−0.195 + 0.564i)13-s + (2.97 + 2.83i)14-s + (−1.80 − 3.35i)15-s + (1.07 + 3.11i)16-s + (−0.498 + 1.09i)17-s + ⋯ |
| L(s) = 1 | + (−0.421 − 0.817i)2-s + (−0.967 + 0.253i)3-s + (0.0888 − 0.124i)4-s + (0.232 + 0.956i)5-s + (0.615 + 0.684i)6-s + (−1.12 + 0.390i)7-s + (−1.05 − 0.151i)8-s + (0.871 − 0.490i)9-s + (0.684 − 0.593i)10-s + (0.0931 + 1.95i)11-s + (−0.0543 + 0.143i)12-s + (−0.0542 + 0.156i)13-s + (0.794 + 0.757i)14-s + (−0.466 − 0.866i)15-s + (0.269 + 0.778i)16-s + (−0.120 + 0.264i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.416 - 0.908i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.416 - 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.420836 + 0.269982i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.420836 + 0.269982i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.67 - 0.439i)T \) |
| 23 | \( 1 + (0.627 - 4.75i)T \) |
| good | 2 | \( 1 + (0.596 + 1.15i)T + (-1.16 + 1.62i)T^{2} \) |
| 5 | \( 1 + (-0.518 - 2.13i)T + (-4.44 + 2.29i)T^{2} \) |
| 7 | \( 1 + (2.98 - 1.03i)T + (5.50 - 4.32i)T^{2} \) |
| 11 | \( 1 + (-0.308 - 6.48i)T + (-10.9 + 1.04i)T^{2} \) |
| 13 | \( 1 + (0.195 - 0.564i)T + (-10.2 - 8.03i)T^{2} \) |
| 17 | \( 1 + (0.498 - 1.09i)T + (-11.1 - 12.8i)T^{2} \) |
| 19 | \( 1 + (-2.29 + 1.04i)T + (12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (-1.09 + 0.781i)T + (9.48 - 27.4i)T^{2} \) |
| 31 | \( 1 + (4.11 - 1.64i)T + (22.4 - 21.3i)T^{2} \) |
| 37 | \( 1 + (1.94 + 6.63i)T + (-31.1 + 20.0i)T^{2} \) |
| 41 | \( 1 + (10.4 - 2.53i)T + (36.4 - 18.7i)T^{2} \) |
| 43 | \( 1 + (0.922 - 2.30i)T + (-31.1 - 29.6i)T^{2} \) |
| 47 | \( 1 + (3.46 + 2.00i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-6.44 + 7.44i)T + (-7.54 - 52.4i)T^{2} \) |
| 59 | \( 1 + (-3.80 - 1.31i)T + (46.3 + 36.4i)T^{2} \) |
| 61 | \( 1 + (1.30 - 1.66i)T + (-14.3 - 59.2i)T^{2} \) |
| 67 | \( 1 + (-6.95 - 0.331i)T + (66.6 + 6.36i)T^{2} \) |
| 71 | \( 1 + (-5.22 + 8.13i)T + (-29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (-6.46 - 14.1i)T + (-47.8 + 55.1i)T^{2} \) |
| 79 | \( 1 + (-0.120 - 0.623i)T + (-73.3 + 29.3i)T^{2} \) |
| 83 | \( 1 + (-1.59 + 6.57i)T + (-73.7 - 38.0i)T^{2} \) |
| 89 | \( 1 + (-0.587 - 4.08i)T + (-85.3 + 25.0i)T^{2} \) |
| 97 | \( 1 + (-13.2 - 13.8i)T + (-4.61 + 96.8i)T^{2} \) |
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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
−12.24927999693278842334541716342, −11.53595116673397864038613334065, −10.48737748507084883527789659016, −9.873004056227255516299574028226, −9.372228048506572719321235885566, −7.03924109533340205189192477721, −6.54774077714650659011675811072, −5.30512660417684990227923320213, −3.53259899083664375235684812314, −2.04167162972955882579859742124,
0.53262445270537143817360192715, 3.38804445092569127941382748556, 5.26558710455709854344895910805, 6.15605864360628166772249803552, 6.89172044671894610213659780557, 8.189178696385937134836001896585, 9.019659783356069500254508686908, 10.18824720698991351694471839643, 11.36205978871463498497848448185, 12.27280850256075241377118731787
