| L(s) = 1 | + (−0.00565 + 0.0265i)2-s + (−1.22 − 0.128i)3-s + (1.82 + 0.813i)4-s + (0.859 − 0.774i)5-s + (0.0103 − 0.0318i)6-s + (2.59 + 0.530i)7-s + (−0.0639 + 0.0879i)8-s + (−1.45 − 0.308i)9-s + (0.0157 + 0.0272i)10-s + (−2.03 + 2.61i)11-s + (−2.13 − 1.23i)12-s + (−1.77 − 5.45i)13-s + (−0.0287 + 0.0659i)14-s + (−1.15 + 0.837i)15-s + (2.67 + 2.96i)16-s + (−6.10 + 1.29i)17-s + ⋯ |
| L(s) = 1 | + (−0.00399 + 0.0188i)2-s + (−0.706 − 0.0742i)3-s + (0.913 + 0.406i)4-s + (0.384 − 0.346i)5-s + (0.00422 − 0.0129i)6-s + (0.979 + 0.200i)7-s + (−0.0225 + 0.0311i)8-s + (−0.483 − 0.102i)9-s + (0.00497 + 0.00861i)10-s + (−0.613 + 0.789i)11-s + (−0.615 − 0.355i)12-s + (−0.491 − 1.51i)13-s + (−0.00768 + 0.0176i)14-s + (−0.297 + 0.216i)15-s + (0.668 + 0.742i)16-s + (−1.48 + 0.314i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 - 0.0795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.945575 + 0.0376932i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.945575 + 0.0376932i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-2.59 - 0.530i)T \) |
| 11 | \( 1 + (2.03 - 2.61i)T \) |
| good | 2 | \( 1 + (0.00565 - 0.0265i)T + (-1.82 - 0.813i)T^{2} \) |
| 3 | \( 1 + (1.22 + 0.128i)T + (2.93 + 0.623i)T^{2} \) |
| 5 | \( 1 + (-0.859 + 0.774i)T + (0.522 - 4.97i)T^{2} \) |
| 13 | \( 1 + (1.77 + 5.45i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (6.10 - 1.29i)T + (15.5 - 6.91i)T^{2} \) |
| 19 | \( 1 + (2.16 - 0.963i)T + (12.7 - 14.1i)T^{2} \) |
| 23 | \( 1 + (-2.75 + 4.77i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (2.64 + 3.63i)T + (-8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (-1.47 - 1.33i)T + (3.24 + 30.8i)T^{2} \) |
| 37 | \( 1 + (0.290 + 2.75i)T + (-36.1 + 7.69i)T^{2} \) |
| 41 | \( 1 + (-1.14 - 0.828i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 - 1.93iT - 43T^{2} \) |
| 47 | \( 1 + (-1.89 - 4.25i)T + (-31.4 + 34.9i)T^{2} \) |
| 53 | \( 1 + (-5.98 + 6.64i)T + (-5.54 - 52.7i)T^{2} \) |
| 59 | \( 1 + (3.59 - 8.07i)T + (-39.4 - 43.8i)T^{2} \) |
| 61 | \( 1 + (1.33 + 1.48i)T + (-6.37 + 60.6i)T^{2} \) |
| 67 | \( 1 + (5.00 + 8.67i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-0.143 + 0.440i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-0.275 - 0.122i)T + (48.8 + 54.2i)T^{2} \) |
| 79 | \( 1 + (1.33 - 6.29i)T + (-72.1 - 32.1i)T^{2} \) |
| 83 | \( 1 + (-1.25 + 3.87i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (-12.9 - 7.49i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (4.66 - 1.51i)T + (78.4 - 57.0i)T^{2} \) |
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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
−14.87304119347171673976653120529, −13.04279151984603042002927678644, −12.32415944401459440497926483653, −11.17234674101377200973279177717, −10.49781042179737262777562272604, −8.618630965097090138412330230842, −7.51107168757060541301061060292, −6.06892345267104587026457378893, −4.92265565255655571212903766184, −2.38706673953245012228513422551,
2.27876867169857727698466427852, 4.89517673686252098768188331146, 6.13163748628342546746913615543, 7.19180764376528801178640715808, 8.840788989026862064642689159522, 10.50125025773222273591957557163, 11.22903131060049200598299764782, 11.75755350433179890091735097900, 13.62526698910827952882534000875, 14.46009689215267725686716646423
