| L(s) = 1 | + (0.290 − 0.0945i)2-s + (0.964 + 0.964i)3-s + (−1.54 + 1.12i)4-s + (−2.03 − 2.80i)5-s + (0.371 + 0.189i)6-s + (1.29 − 0.661i)7-s + (−0.702 + 0.966i)8-s − 1.13i·9-s + (−0.857 − 0.623i)10-s + (0.285 + 1.80i)11-s + (−2.56 − 0.406i)12-s + (−2.91 + 5.71i)13-s + (0.314 − 0.314i)14-s + (0.739 − 4.67i)15-s + (1.06 − 3.27i)16-s + (3.14 − 0.497i)17-s + ⋯ |
| L(s) = 1 | + (0.205 − 0.0668i)2-s + (0.557 + 0.557i)3-s + (−0.771 + 0.560i)4-s + (−0.911 − 1.25i)5-s + (0.151 + 0.0773i)6-s + (0.490 − 0.249i)7-s + (−0.248 + 0.341i)8-s − 0.379i·9-s + (−0.271 − 0.197i)10-s + (0.0861 + 0.544i)11-s + (−0.741 − 0.117i)12-s + (−0.807 + 1.58i)13-s + (0.0841 − 0.0841i)14-s + (0.191 − 1.20i)15-s + (0.266 − 0.819i)16-s + (0.762 − 0.120i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.182i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.983 - 0.182i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.799188 + 0.0736237i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.799188 + 0.0736237i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 41 | \( 1 + (6.22 - 1.51i)T \) |
| good | 2 | \( 1 + (-0.290 + 0.0945i)T + (1.61 - 1.17i)T^{2} \) |
| 3 | \( 1 + (-0.964 - 0.964i)T + 3iT^{2} \) |
| 5 | \( 1 + (2.03 + 2.80i)T + (-1.54 + 4.75i)T^{2} \) |
| 7 | \( 1 + (-1.29 + 0.661i)T + (4.11 - 5.66i)T^{2} \) |
| 11 | \( 1 + (-0.285 - 1.80i)T + (-10.4 + 3.39i)T^{2} \) |
| 13 | \( 1 + (2.91 - 5.71i)T + (-7.64 - 10.5i)T^{2} \) |
| 17 | \( 1 + (-3.14 + 0.497i)T + (16.1 - 5.25i)T^{2} \) |
| 19 | \( 1 + (-1.51 - 2.98i)T + (-11.1 + 15.3i)T^{2} \) |
| 23 | \( 1 + (1.67 + 5.15i)T + (-18.6 + 13.5i)T^{2} \) |
| 29 | \( 1 + (0.335 + 0.0530i)T + (27.5 + 8.96i)T^{2} \) |
| 31 | \( 1 + (3.04 + 2.21i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-0.993 + 0.721i)T + (11.4 - 35.1i)T^{2} \) |
| 43 | \( 1 + (-7.02 + 2.28i)T + (34.7 - 25.2i)T^{2} \) |
| 47 | \( 1 + (1.40 + 0.717i)T + (27.6 + 38.0i)T^{2} \) |
| 53 | \( 1 + (-1.12 - 0.178i)T + (50.4 + 16.3i)T^{2} \) |
| 59 | \( 1 + (-4.16 - 12.8i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (4.39 + 1.42i)T + (49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (-1.04 + 6.58i)T + (-63.7 - 20.7i)T^{2} \) |
| 71 | \( 1 + (1.29 + 8.16i)T + (-67.5 + 21.9i)T^{2} \) |
| 73 | \( 1 + 8.99iT - 73T^{2} \) |
| 79 | \( 1 + (-7.20 - 7.20i)T + 79iT^{2} \) |
| 83 | \( 1 + 7.11T + 83T^{2} \) |
| 89 | \( 1 + (12.9 - 6.58i)T + (52.3 - 72.0i)T^{2} \) |
| 97 | \( 1 + (-0.514 + 3.24i)T + (-92.2 - 29.9i)T^{2} \) |
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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
−16.35718320787444544963408770968, −14.83876010608162339778928759885, −14.02797964916027193478806355748, −12.39330839888426721986365529845, −11.94477543844490796994920713949, −9.627435336027962257019689048024, −8.778987479715076207307689172764, −7.67604470189068311186888821280, −4.71155629302214923788379004155, −3.98509778115369306253567560798,
3.22065244054584463765262350270, 5.38073270830049420991210519006, 7.36230037519773535495909494431, 8.280730262326684229937125019640, 10.11166671937126021067085249857, 11.27831929959631091630977416697, 12.81975898158854013115624991924, 14.04601493404587901300190540581, 14.74343201938340353480726835450, 15.61014245238774899385141859730
