L(s) = 1 | + (−0.415 + 0.909i)3-s + (−0.654 + 0.755i)5-s + (−0.841 + 0.540i)7-s + (−0.654 − 0.755i)9-s + (0.142 − 0.989i)11-s + (0.841 + 0.540i)13-s + (−0.415 − 0.909i)15-s + (−0.959 − 0.281i)17-s + (0.959 − 0.281i)19-s + (−0.142 − 0.989i)21-s + (−0.142 − 0.989i)25-s + (0.959 − 0.281i)27-s + (−0.959 − 0.281i)29-s + (−0.415 − 0.909i)31-s + (0.841 + 0.540i)33-s + ⋯ |
L(s) = 1 | + (−0.415 + 0.909i)3-s + (−0.654 + 0.755i)5-s + (−0.841 + 0.540i)7-s + (−0.654 − 0.755i)9-s + (0.142 − 0.989i)11-s + (0.841 + 0.540i)13-s + (−0.415 − 0.909i)15-s + (−0.959 − 0.281i)17-s + (0.959 − 0.281i)19-s + (−0.142 − 0.989i)21-s + (−0.142 − 0.989i)25-s + (0.959 − 0.281i)27-s + (−0.959 − 0.281i)29-s + (−0.415 − 0.909i)31-s + (0.841 + 0.540i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.293 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.293 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.03205446198 - 0.04334947363i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.03205446198 - 0.04334947363i\) |
\(L(1)\) |
\(\approx\) |
\(0.5672425983 + 0.2168791443i\) |
\(L(1)\) |
\(\approx\) |
\(0.5672425983 + 0.2168791443i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 23 | \( 1 \) |
good | 3 | \( 1 + (-0.415 + 0.909i)T \) |
| 5 | \( 1 + (-0.654 + 0.755i)T \) |
| 7 | \( 1 + (-0.841 + 0.540i)T \) |
| 11 | \( 1 + (0.142 - 0.989i)T \) |
| 13 | \( 1 + (0.841 + 0.540i)T \) |
| 17 | \( 1 + (-0.959 - 0.281i)T \) |
| 19 | \( 1 + (0.959 - 0.281i)T \) |
| 29 | \( 1 + (-0.959 - 0.281i)T \) |
| 31 | \( 1 + (-0.415 - 0.909i)T \) |
| 37 | \( 1 + (-0.654 - 0.755i)T \) |
| 41 | \( 1 + (-0.654 + 0.755i)T \) |
| 43 | \( 1 + (-0.415 + 0.909i)T \) |
| 47 | \( 1 - T \) |
| 53 | \( 1 + (0.841 - 0.540i)T \) |
| 59 | \( 1 + (-0.841 - 0.540i)T \) |
| 61 | \( 1 + (0.415 + 0.909i)T \) |
| 67 | \( 1 + (0.142 + 0.989i)T \) |
| 71 | \( 1 + (0.142 + 0.989i)T \) |
| 73 | \( 1 + (-0.959 + 0.281i)T \) |
| 79 | \( 1 + (-0.841 - 0.540i)T \) |
| 83 | \( 1 + (0.654 + 0.755i)T \) |
| 89 | \( 1 + (0.415 - 0.909i)T \) |
| 97 | \( 1 + (-0.654 + 0.755i)T \) |
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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
−30.53273715745721181760654698313, −29.204784730390528400765976019166, −28.4932677616779296306365880177, −27.540502325081611666163510594766, −26.04575126863347457964098115264, −25.03762262745029541480491676139, −24.000985795523149731311371438480, −23.093763789451482768103706645371, −22.41130366954420493217085035257, −20.336342152685759454613026465808, −19.90269483475767145419761186946, −18.631339304605737655002903322989, −17.489360596258954437873666941311, −16.47417208806888063402178777919, −15.43762529017711407902316535888, −13.609214121069361117917912825517, −12.812825096627019922328254096984, −11.91118730720346920180904774764, −10.575644936459644887873465641881, −8.955719779541627995346621707078, −7.6460874684276345218335422520, −6.66150727616846477276422084060, −5.16778944999664416858687989329, −3.586343047280773912186090718401, −1.47065358511391874125696918336,
0.02666756667046684328547814893, 3.0234035064113433884954816974, 3.97608070231404472150270462634, 5.72846130675755650232921954377, 6.76035741145408786622345816478, 8.627390547527390057043442842458, 9.690469839225809713611447933875, 11.1338562455458090370177393947, 11.62612769056759329321838670279, 13.414593160753678873666088244941, 14.8093490553990374700818800893, 15.85992877714988686009452716042, 16.386473589331503213636175939929, 18.0753852777969686956855441839, 19.03873654024660254929937801529, 20.206021267896913765246048634074, 21.617181898221299098329583017704, 22.33206895520508819745119385713, 23.134713518296406579482894272737, 24.43687397181695660952779532195, 26.13502610550270559477679405449, 26.49116799978401800905189225479, 27.700566412867025369837643964358, 28.63279954212661400458995045577, 29.608544481102853695188230230434