| 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
−14.46009689215267725686716646423, −13.62526698910827952882534000875, −11.75755350433179890091735097900, −11.22903131060049200598299764782, −10.50125025773222273591957557163, −8.840788989026862064642689159522, −7.19180764376528801178640715808, −6.13163748628342546746913615543, −4.89517673686252098768188331146, −2.27876867169857727698466427852,
2.38706673953245012228513422551, 4.92265565255655571212903766184, 6.06892345267104587026457378893, 7.51107168757060541301061060292, 8.618630965097090138412330230842, 10.49781042179737262777562272604, 11.17234674101377200973279177717, 12.32415944401459440497926483653, 13.04279151984603042002927678644, 14.87304119347171673976653120529
