| L(s) = 1 | + (−0.507 + 0.254i)2-s + (1.73 + 0.0683i)3-s + (−1.00 + 1.34i)4-s + (−0.832 + 2.78i)5-s + (−0.895 + 0.406i)6-s + (1.23 − 2.87i)7-s + (0.362 − 2.05i)8-s + (2.99 + 0.236i)9-s + (−0.286 − 1.62i)10-s + (−3.33 − 0.791i)11-s + (−1.82 + 2.26i)12-s + (3.33 − 2.19i)13-s + (0.103 + 1.77i)14-s + (−1.63 + 4.75i)15-s + (−0.622 − 2.07i)16-s + (0.878 + 0.319i)17-s + ⋯ |
| L(s) = 1 | + (−0.358 + 0.180i)2-s + (0.999 + 0.0394i)3-s + (−0.500 + 0.672i)4-s + (−0.372 + 1.24i)5-s + (−0.365 + 0.165i)6-s + (0.468 − 1.08i)7-s + (0.128 − 0.726i)8-s + (0.996 + 0.0788i)9-s + (−0.0905 − 0.513i)10-s + (−1.00 − 0.238i)11-s + (−0.527 + 0.652i)12-s + (0.924 − 0.607i)13-s + (0.0276 + 0.473i)14-s + (−0.421 + 1.22i)15-s + (−0.155 − 0.519i)16-s + (0.213 + 0.0775i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.688 - 0.725i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.688 - 0.725i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.868742 + 0.373361i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.868742 + 0.373361i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.73 - 0.0683i)T \) |
| good | 2 | \( 1 + (0.507 - 0.254i)T + (1.19 - 1.60i)T^{2} \) |
| 5 | \( 1 + (0.832 - 2.78i)T + (-4.17 - 2.74i)T^{2} \) |
| 7 | \( 1 + (-1.23 + 2.87i)T + (-4.80 - 5.09i)T^{2} \) |
| 11 | \( 1 + (3.33 + 0.791i)T + (9.82 + 4.93i)T^{2} \) |
| 13 | \( 1 + (-3.33 + 2.19i)T + (5.14 - 11.9i)T^{2} \) |
| 17 | \( 1 + (-0.878 - 0.319i)T + (13.0 + 10.9i)T^{2} \) |
| 19 | \( 1 + (4.55 - 1.65i)T + (14.5 - 12.2i)T^{2} \) |
| 23 | \( 1 + (2.43 + 5.64i)T + (-15.7 + 16.7i)T^{2} \) |
| 29 | \( 1 + (0.104 - 1.79i)T + (-28.8 - 3.36i)T^{2} \) |
| 31 | \( 1 + (0.671 + 0.0784i)T + (30.1 + 7.14i)T^{2} \) |
| 37 | \( 1 + (8.73 - 7.32i)T + (6.42 - 36.4i)T^{2} \) |
| 41 | \( 1 + (-5.94 - 2.98i)T + (24.4 + 32.8i)T^{2} \) |
| 43 | \( 1 + (-1.53 + 1.62i)T + (-2.50 - 42.9i)T^{2} \) |
| 47 | \( 1 + (-3.14 + 0.367i)T + (45.7 - 10.8i)T^{2} \) |
| 53 | \( 1 + (4.18 + 7.25i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (4.89 - 1.16i)T + (52.7 - 26.4i)T^{2} \) |
| 61 | \( 1 + (0.340 + 0.457i)T + (-17.4 + 58.4i)T^{2} \) |
| 67 | \( 1 + (-0.794 - 13.6i)T + (-66.5 + 7.77i)T^{2} \) |
| 71 | \( 1 + (2.31 + 13.1i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (1.17 - 6.64i)T + (-68.5 - 24.9i)T^{2} \) |
| 79 | \( 1 + (-0.666 + 0.334i)T + (47.1 - 63.3i)T^{2} \) |
| 83 | \( 1 + (3.25 - 1.63i)T + (49.5 - 66.5i)T^{2} \) |
| 89 | \( 1 + (2.27 - 12.9i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (3.58 + 11.9i)T + (-81.0 + 53.3i)T^{2} \) |
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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
−14.40534443598561427580050130212, −13.62196621387789113070561120230, −12.68236899196842500253338481839, −10.76026733976866165761593062500, −10.24653495917578240371598715517, −8.451555810864138830626951916465, −7.87355857072363576164356233834, −6.86044388330376558901926628969, −4.18157658659941505368474044736, −3.10803761860740475084359717354,
1.89925665093901695527591136577, 4.36909039105530218872111695454, 5.57081781672484753712152564024, 7.929706920827454041300377642459, 8.781857449731916000591414828246, 9.292809172117708244746167422256, 10.78134579199324001523589947518, 12.25239272446549410397850217496, 13.21615604066376868431211340283, 14.18045373379469124514129069454
