| 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
−14.18045373379469124514129069454, −13.21615604066376868431211340283, −12.25239272446549410397850217496, −10.78134579199324001523589947518, −9.292809172117708244746167422256, −8.781857449731916000591414828246, −7.929706920827454041300377642459, −5.57081781672484753712152564024, −4.36909039105530218872111695454, −1.89925665093901695527591136577,
3.10803761860740475084359717354, 4.18157658659941505368474044736, 6.86044388330376558901926628969, 7.87355857072363576164356233834, 8.451555810864138830626951916465, 10.24653495917578240371598715517, 10.76026733976866165761593062500, 12.68236899196842500253338481839, 13.62196621387789113070561120230, 14.40534443598561427580050130212
