L(s) = 1 | + (−0.148 + 0.988i)2-s + (−0.627 + 0.778i)3-s + (−0.956 − 0.293i)4-s + (0.945 − 0.324i)5-s + (−0.677 − 0.735i)6-s + (0.309 + 0.951i)7-s + (0.431 − 0.901i)8-s + (−0.213 − 0.976i)9-s + (0.180 + 0.983i)10-s + (0.245 + 0.969i)11-s + (0.828 − 0.560i)12-s + (0.0495 + 0.998i)13-s + (−0.986 + 0.164i)14-s + (−0.340 + 0.940i)15-s + (0.828 + 0.560i)16-s + (−0.518 + 0.854i)17-s + ⋯ |
L(s) = 1 | + (−0.148 + 0.988i)2-s + (−0.627 + 0.778i)3-s + (−0.956 − 0.293i)4-s + (0.945 − 0.324i)5-s + (−0.677 − 0.735i)6-s + (0.309 + 0.951i)7-s + (0.431 − 0.901i)8-s + (−0.213 − 0.976i)9-s + (0.180 + 0.983i)10-s + (0.245 + 0.969i)11-s + (0.828 − 0.560i)12-s + (0.0495 + 0.998i)13-s + (−0.986 + 0.164i)14-s + (−0.340 + 0.940i)15-s + (0.828 + 0.560i)16-s + (−0.518 + 0.854i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 191 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.943 + 0.330i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 191 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.943 + 0.330i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1456703475 + 0.8573514326i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1456703475 + 0.8573514326i\) |
\(L(1)\) |
\(\approx\) |
\(0.5591342394 + 0.6486539516i\) |
\(L(1)\) |
\(\approx\) |
\(0.5591342394 + 0.6486539516i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 191 | \( 1 \) |
good | 2 | \( 1 + (-0.148 + 0.988i)T \) |
| 3 | \( 1 + (-0.627 + 0.778i)T \) |
| 5 | \( 1 + (0.945 - 0.324i)T \) |
| 7 | \( 1 + (0.309 + 0.951i)T \) |
| 11 | \( 1 + (0.245 + 0.969i)T \) |
| 13 | \( 1 + (0.0495 + 0.998i)T \) |
| 17 | \( 1 + (-0.518 + 0.854i)T \) |
| 19 | \( 1 + (0.180 - 0.983i)T \) |
| 23 | \( 1 + (-0.724 + 0.689i)T \) |
| 29 | \( 1 + (-0.277 - 0.960i)T \) |
| 31 | \( 1 + (-0.401 + 0.915i)T \) |
| 37 | \( 1 + (-0.401 - 0.915i)T \) |
| 41 | \( 1 + (-0.986 - 0.164i)T \) |
| 43 | \( 1 + (0.115 + 0.993i)T \) |
| 47 | \( 1 + (0.371 - 0.928i)T \) |
| 53 | \( 1 + (0.997 + 0.0660i)T \) |
| 59 | \( 1 + (0.746 + 0.665i)T \) |
| 61 | \( 1 + (-0.973 + 0.229i)T \) |
| 67 | \( 1 + (0.652 - 0.757i)T \) |
| 71 | \( 1 + (0.701 + 0.712i)T \) |
| 73 | \( 1 + (-0.768 + 0.639i)T \) |
| 79 | \( 1 + (-0.277 + 0.960i)T \) |
| 83 | \( 1 + (-0.724 - 0.689i)T \) |
| 89 | \( 1 + (0.980 - 0.197i)T \) |
| 97 | \( 1 + (0.863 + 0.504i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−26.9669176801849278712253342410, −25.7988925163539414615069845781, −24.66848538810235120910536014373, −23.68489965445824992623318471092, −22.45416731304255877218166631433, −22.189816612426366139030494335006, −20.74767449593895879370645713134, −20.06518010598279898382529376469, −18.70308643219482053218710164795, −18.19243582800368374683780478699, −17.249346128366455970054352689618, −16.55748154696919314741516664630, −14.27896492319825843005915715136, −13.66833268304337165556379656535, −12.89032252042167091733578666030, −11.66359173697260584437835950210, −10.734150693156529567262134821725, −10.11162373606000054856827347105, −8.570030715027894778771741891018, −7.43529116994956992324894624002, −6.0604897980547182432835555085, −5.01352221527557979324705171741, −3.3647244700921044090117343128, −2.00471590394842086434248420347, −0.81944602898365185413935852722,
1.84706993876375333119273083432, 4.16904324966172959198071243676, 5.09780458001861913443145962438, 5.94748166199854817424538708095, 6.89959620965925289255556084820, 8.73686174386716445820710372207, 9.31331316188532116461417867773, 10.22272694039778629458813304655, 11.70580838767377665936324541623, 12.85103448900516403639340057134, 14.12841943479581520408877016766, 15.09285892876952116663511816168, 15.82641253377497411208513989230, 16.95155146106454959306068462529, 17.61803428945090239392897632637, 18.27540885820169352127891594728, 19.82636267886651121644212340436, 21.44135367671495686920904750199, 21.70139996176092531401275866934, 22.74738026662641321495159992105, 23.88316536918084222117948477754, 24.64973793206770466537638408594, 25.77768886155983512290650982397, 26.282726504891881755711365713798, 27.598385168748329204525572849147