L(s) = 1 | + (0.984 + 0.173i)2-s + (0.642 − 0.766i)3-s + (0.939 + 0.342i)4-s + (0.766 − 0.642i)6-s + (−0.866 + 0.5i)7-s + (0.866 + 0.5i)8-s + (−0.173 − 0.984i)9-s + (−0.5 + 0.866i)11-s + (0.866 − 0.5i)12-s + (−0.642 − 0.766i)13-s + (−0.939 + 0.342i)14-s + (0.766 + 0.642i)16-s + (−0.984 − 0.173i)17-s − i·18-s + (−0.173 + 0.984i)21-s + (−0.642 + 0.766i)22-s + ⋯ |
L(s) = 1 | + (0.984 + 0.173i)2-s + (0.642 − 0.766i)3-s + (0.939 + 0.342i)4-s + (0.766 − 0.642i)6-s + (−0.866 + 0.5i)7-s + (0.866 + 0.5i)8-s + (−0.173 − 0.984i)9-s + (−0.5 + 0.866i)11-s + (0.866 − 0.5i)12-s + (−0.642 − 0.766i)13-s + (−0.939 + 0.342i)14-s + (0.766 + 0.642i)16-s + (−0.984 − 0.173i)17-s − i·18-s + (−0.173 + 0.984i)21-s + (−0.642 + 0.766i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.979 - 0.203i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.979 - 0.203i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.872581779 - 0.1927671973i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.872581779 - 0.1927671973i\) |
\(L(1)\) |
\(\approx\) |
\(1.838692773 - 0.1274764926i\) |
\(L(1)\) |
\(\approx\) |
\(1.838692773 - 0.1274764926i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + (0.984 + 0.173i)T \) |
| 3 | \( 1 + (0.642 - 0.766i)T \) |
| 7 | \( 1 + (-0.866 + 0.5i)T \) |
| 11 | \( 1 + (-0.5 + 0.866i)T \) |
| 13 | \( 1 + (-0.642 - 0.766i)T \) |
| 17 | \( 1 + (-0.984 - 0.173i)T \) |
| 23 | \( 1 + (0.342 - 0.939i)T \) |
| 29 | \( 1 + (0.173 + 0.984i)T \) |
| 31 | \( 1 + (0.5 + 0.866i)T \) |
| 37 | \( 1 + iT \) |
| 41 | \( 1 + (-0.766 - 0.642i)T \) |
| 43 | \( 1 + (-0.342 - 0.939i)T \) |
| 47 | \( 1 + (0.984 - 0.173i)T \) |
| 53 | \( 1 + (-0.342 + 0.939i)T \) |
| 59 | \( 1 + (0.173 - 0.984i)T \) |
| 61 | \( 1 + (-0.939 - 0.342i)T \) |
| 67 | \( 1 + (-0.984 + 0.173i)T \) |
| 71 | \( 1 + (0.939 - 0.342i)T \) |
| 73 | \( 1 + (-0.642 + 0.766i)T \) |
| 79 | \( 1 + (0.766 + 0.642i)T \) |
| 83 | \( 1 + (0.866 - 0.5i)T \) |
| 89 | \( 1 + (0.766 - 0.642i)T \) |
| 97 | \( 1 + (0.984 + 0.173i)T \) |
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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
−30.41029597575508296776373213810, −29.28186867090332138439439688454, −28.467702320017509352337355439883, −26.819406438470051887779239680997, −26.12392311403299272096626068332, −24.92929196444987797893796090577, −23.867302513918903462714815616214, −22.64997420602986967580393228094, −21.75080390713193268289611742871, −20.96027158433039401103969251877, −19.7263724250016436163428366167, −19.19235863796188257490433810291, −16.84281317016328924991155021665, −15.967682138243349710038256569416, −15.057810853994312455302728401749, −13.762826536745675255444201411904, −13.21883543704660395829562328940, −11.513002548531072821642746925, −10.42352110687838852829670602370, −9.32348745106667839689336299318, −7.60210414666074883314035644252, −6.164239739083310320957733936968, −4.668377472672580618799167832060, −3.59733237669637209188322570108, −2.442516123299454621555440937951,
2.253504331931098602562747865621, 3.17684486524748081578099920894, 4.93849685011049851135768831656, 6.45675950552679809155241353225, 7.30656765843295031237701236702, 8.70776911578193548721151721662, 10.33157581797586164342841406784, 12.18880689077876370797001311560, 12.710030576532295790227537754759, 13.711591672133202478429149178314, 15.00144156952645036504489420355, 15.66199285377159154856810057032, 17.28551998311209163670323973528, 18.58472984312856902353406358816, 19.88427169479194218975888137401, 20.46465511416569667537890581697, 21.97360888663577720809084172648, 22.856079740098656328042335457124, 23.90651225877103055645157690669, 25.00675715110188857065329376024, 25.51403657497808358996520939323, 26.632453224058986899377035753760, 28.65165098613031230086737917449, 29.2822798264595264976187778301, 30.46198814074760048596213738356