L(s) = 1 | + (−0.969 − 0.244i)2-s + (0.879 + 0.475i)4-s + (−0.861 + 0.508i)5-s + (−0.161 + 0.986i)7-s + (−0.736 − 0.676i)8-s + (0.959 − 0.281i)10-s + (0.991 − 0.132i)13-s + (0.398 − 0.917i)14-s + (0.548 + 0.836i)16-s + (0.0855 − 0.996i)17-s + (0.921 − 0.389i)19-s + (−0.999 + 0.0380i)20-s + (0.888 − 0.458i)23-s + (0.483 − 0.875i)25-s + (−0.993 − 0.113i)26-s + ⋯ |
L(s) = 1 | + (−0.969 − 0.244i)2-s + (0.879 + 0.475i)4-s + (−0.861 + 0.508i)5-s + (−0.161 + 0.986i)7-s + (−0.736 − 0.676i)8-s + (0.959 − 0.281i)10-s + (0.991 − 0.132i)13-s + (0.398 − 0.917i)14-s + (0.548 + 0.836i)16-s + (0.0855 − 0.996i)17-s + (0.921 − 0.389i)19-s + (−0.999 + 0.0380i)20-s + (0.888 − 0.458i)23-s + (0.483 − 0.875i)25-s + (−0.993 − 0.113i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.941 - 0.336i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.941 - 0.336i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7485401890 - 0.1297623983i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7485401890 - 0.1297623983i\) |
\(L(1)\) |
\(\approx\) |
\(0.6444372710 + 0.003567305562i\) |
\(L(1)\) |
\(\approx\) |
\(0.6444372710 + 0.003567305562i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.969 - 0.244i)T \) |
| 5 | \( 1 + (-0.861 + 0.508i)T \) |
| 7 | \( 1 + (-0.161 + 0.986i)T \) |
| 13 | \( 1 + (0.991 - 0.132i)T \) |
| 17 | \( 1 + (0.0855 - 0.996i)T \) |
| 19 | \( 1 + (0.921 - 0.389i)T \) |
| 23 | \( 1 + (0.888 - 0.458i)T \) |
| 29 | \( 1 + (-0.999 - 0.0190i)T \) |
| 31 | \( 1 + (-0.761 + 0.647i)T \) |
| 37 | \( 1 + (-0.564 - 0.825i)T \) |
| 41 | \( 1 + (-0.830 + 0.556i)T \) |
| 43 | \( 1 + (0.995 - 0.0950i)T \) |
| 47 | \( 1 + (-0.345 - 0.938i)T \) |
| 53 | \( 1 + (0.998 + 0.0570i)T \) |
| 59 | \( 1 + (0.0665 - 0.997i)T \) |
| 61 | \( 1 + (-0.272 + 0.962i)T \) |
| 67 | \( 1 + (-0.786 - 0.618i)T \) |
| 71 | \( 1 + (-0.974 - 0.226i)T \) |
| 73 | \( 1 + (0.254 - 0.967i)T \) |
| 79 | \( 1 + (0.948 - 0.318i)T \) |
| 83 | \( 1 + (0.988 - 0.151i)T \) |
| 89 | \( 1 + (0.654 - 0.755i)T \) |
| 97 | \( 1 + (0.861 + 0.508i)T \) |
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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
−20.92739776025209919012380401581, −20.60306173034512428266291194515, −19.82584010746969655224446794468, −19.13028422614878150866860414513, −18.50817290753137682914606175134, −17.427737110526002027748010642952, −16.786919476106274769982047346407, −16.22781479661553203398933140490, −15.45793210965484765277745459697, −14.71631677110910123144358932173, −13.602550554288401034392847666056, −12.764883353200363243176388676511, −11.69238627665985964487234401156, −11.05305740827543548863110072365, −10.35189570367299511172019313701, −9.32546124948812738340013096551, −8.62113660755213855639818106787, −7.71529163588799649089716527031, −7.26104051007441159716385890624, −6.19545277115070375359011694231, −5.236210149718542027486112245470, −3.935434207393000206797267609115, −3.314930922205073837643298013671, −1.61076252701639656248045661341, −0.893626811909202466025843030570,
0.60379794244987847497264159525, 1.98037027242530043830129248982, 3.05654975978979193180697532449, 3.527695016220296634712835730957, 5.06728952064583217099902191130, 6.14082122039052443836494642316, 7.08899183059245488399965790537, 7.67286993275605803772499236086, 8.82555083923867613875796945605, 9.086544091253334233532440097657, 10.30943466389707770010428918122, 11.12685355584252071831128604187, 11.671798581371315750584331684816, 12.38089309861979063784474311301, 13.37545129209141197469670889000, 14.67523466837825908612490812327, 15.37737369682978175932456626289, 16.03455498961893549456960716748, 16.55384511945342108328755609254, 17.9842246123162850198899962430, 18.28238421759799288987455650181, 18.942532614360572670387546637545, 19.68692477566846084757184116720, 20.48977217791160611774884823604, 21.16167905646704931956238087156