| L(s) = 1 | + (2.72 − 0.410i)2-s + (0.680 + 0.463i)3-s + (5.34 − 1.64i)4-s + (0.189 − 2.52i)5-s + (2.04 + 0.984i)6-s + (−1.91 + 1.82i)7-s + (8.92 − 4.29i)8-s + (−0.848 − 2.16i)9-s + (−0.522 − 6.96i)10-s + (−2.37 + 6.04i)11-s + (4.40 + 1.35i)12-s + (−0.623 − 0.781i)13-s + (−4.47 + 5.75i)14-s + (1.30 − 1.63i)15-s + (13.3 − 9.07i)16-s + (−3.06 + 2.84i)17-s + ⋯ |
| L(s) = 1 | + (1.92 − 0.290i)2-s + (0.392 + 0.267i)3-s + (2.67 − 0.824i)4-s + (0.0847 − 1.13i)5-s + (0.834 + 0.401i)6-s + (−0.724 + 0.689i)7-s + (3.15 − 1.51i)8-s + (−0.282 − 0.720i)9-s + (−0.165 − 2.20i)10-s + (−0.714 + 1.82i)11-s + (1.27 + 0.391i)12-s + (−0.172 − 0.216i)13-s + (−1.19 + 1.53i)14-s + (0.336 − 0.421i)15-s + (3.32 − 2.26i)16-s + (−0.743 + 0.690i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.827 + 0.560i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.827 + 0.560i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.58604 - 1.40688i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.58604 - 1.40688i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (1.91 - 1.82i)T \) |
| 13 | \( 1 + (0.623 + 0.781i)T \) |
| good | 2 | \( 1 + (-2.72 + 0.410i)T + (1.91 - 0.589i)T^{2} \) |
| 3 | \( 1 + (-0.680 - 0.463i)T + (1.09 + 2.79i)T^{2} \) |
| 5 | \( 1 + (-0.189 + 2.52i)T + (-4.94 - 0.745i)T^{2} \) |
| 11 | \( 1 + (2.37 - 6.04i)T + (-8.06 - 7.48i)T^{2} \) |
| 17 | \( 1 + (3.06 - 2.84i)T + (1.27 - 16.9i)T^{2} \) |
| 19 | \( 1 + (0.335 - 0.581i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.258 - 0.240i)T + (1.71 + 22.9i)T^{2} \) |
| 29 | \( 1 + (-1.18 - 5.20i)T + (-26.1 + 12.5i)T^{2} \) |
| 31 | \( 1 + (3.77 + 6.53i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.53 - 0.780i)T + (30.5 + 20.8i)T^{2} \) |
| 41 | \( 1 + (7.57 - 3.64i)T + (25.5 - 32.0i)T^{2} \) |
| 43 | \( 1 + (-5.72 - 2.75i)T + (26.8 + 33.6i)T^{2} \) |
| 47 | \( 1 + (-9.21 + 1.38i)T + (44.9 - 13.8i)T^{2} \) |
| 53 | \( 1 + (0.141 - 0.0435i)T + (43.7 - 29.8i)T^{2} \) |
| 59 | \( 1 + (-0.485 - 6.48i)T + (-58.3 + 8.79i)T^{2} \) |
| 61 | \( 1 + (-0.758 - 0.234i)T + (50.4 + 34.3i)T^{2} \) |
| 67 | \( 1 + (4.66 + 8.08i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-2.58 + 11.3i)T + (-63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (12.3 + 1.85i)T + (69.7 + 21.5i)T^{2} \) |
| 79 | \( 1 + (2.18 - 3.79i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-8.55 + 10.7i)T + (-18.4 - 80.9i)T^{2} \) |
| 89 | \( 1 + (0.924 + 2.35i)T + (-65.2 + 60.5i)T^{2} \) |
| 97 | \( 1 + 5.30T + 97T^{2} \) |
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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
−10.63931245040319295479742941305, −9.784267924537342082541685848763, −8.942118367684949657182352524963, −7.56132856376664180164854256659, −6.50919327371655708427266101704, −5.64909345885626705592575953887, −4.78756545327129116780197504205, −4.08572415515277244808071962623, −2.90242099447078192655728579915, −1.93027045814201924171357138322,
2.54261561538919337292404238185, 2.98314379701685450860505488118, 3.93739449928829693466302693572, 5.24529246577546412438230326815, 6.08383616059343262906879236563, 6.93281585864492040858316668512, 7.45817363142032949785067690608, 8.557638225555027481765918007569, 10.42716710971485633501339241138, 10.91091923103680330565006052499
