| 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
−12.27280850256075241377118731787, −11.36205978871463498497848448185, −10.18824720698991351694471839643, −9.019659783356069500254508686908, −8.189178696385937134836001896585, −6.89172044671894610213659780557, −6.15605864360628166772249803552, −5.26558710455709854344895910805, −3.38804445092569127941382748556, −0.53262445270537143817360192715,
2.04167162972955882579859742124, 3.53259899083664375235684812314, 5.30512660417684990227923320213, 6.54774077714650659011675811072, 7.03924109533340205189192477721, 9.372228048506572719321235885566, 9.873004056227255516299574028226, 10.48737748507084883527789659016, 11.53595116673397864038613334065, 12.24927999693278842334541716342
