L(s) = 1 | + (0.5 + 0.866i)2-s − 3-s + (−0.5 + 0.866i)4-s + (−0.5 + 0.866i)5-s + (−0.5 − 0.866i)6-s − 8-s + 9-s − 10-s − 11-s + (0.5 − 0.866i)12-s + (0.5 − 0.866i)15-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + (0.5 + 0.866i)18-s + 19-s + (−0.5 − 0.866i)20-s + ⋯ |
L(s) = 1 | + (0.5 + 0.866i)2-s − 3-s + (−0.5 + 0.866i)4-s + (−0.5 + 0.866i)5-s + (−0.5 − 0.866i)6-s − 8-s + 9-s − 10-s − 11-s + (0.5 − 0.866i)12-s + (0.5 − 0.866i)15-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + (0.5 + 0.866i)18-s + 19-s + (−0.5 − 0.866i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.113 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.113 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.03690587958 + 0.03292383618i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.03690587958 + 0.03292383618i\) |
\(L(1)\) |
\(\approx\) |
\(0.5292775333 + 0.3920383368i\) |
\(L(1)\) |
\(\approx\) |
\(0.5292775333 + 0.3920383368i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 + (0.5 - 0.866i)T \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 + (-0.5 - 0.866i)T \) |
| 29 | \( 1 + (-0.5 + 0.866i)T \) |
| 31 | \( 1 + (-0.5 - 0.866i)T \) |
| 37 | \( 1 + (0.5 + 0.866i)T \) |
| 41 | \( 1 + (-0.5 + 0.866i)T \) |
| 43 | \( 1 + (-0.5 - 0.866i)T \) |
| 47 | \( 1 + (-0.5 + 0.866i)T \) |
| 53 | \( 1 + (-0.5 - 0.866i)T \) |
| 59 | \( 1 + (-0.5 + 0.866i)T \) |
| 61 | \( 1 - T \) |
| 67 | \( 1 - T \) |
| 71 | \( 1 + (0.5 + 0.866i)T \) |
| 73 | \( 1 + (-0.5 - 0.866i)T \) |
| 79 | \( 1 + (-0.5 + 0.866i)T \) |
| 83 | \( 1 + T \) |
| 89 | \( 1 + (-0.5 - 0.866i)T \) |
| 97 | \( 1 + (-0.5 - 0.866i)T \) |
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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
−29.16107286045576698404465053973, −28.42937220469573332660184061406, −27.775297805722295640459749074437, −26.639048429985379291754807526453, −24.604579912756975848775445726926, −23.659215136921814155252962823282, −23.134989484114658795527943198890, −21.80228868209287068530717676560, −20.993588563770209546182644473706, −19.86239273525843448580467292497, −18.69076682928060915672528934482, −17.619547169384583055662006636544, −16.24851741594278671203144427637, −15.28622437516863153614186852465, −13.48620059791671230853499119415, −12.56107895857648874710287768474, −11.74379720687314163472326933332, −10.64781797775942713110043727080, −9.498716921520984122933475000947, −7.76996382707392668740752133156, −5.787156274838776957966996972034, −4.97410524842875395822203753926, −3.680639668364564465362900767529, −1.50210023976149264941858508070, −0.02250401116457953775278419475,
3.11451161786997496841547581693, 4.66497683571892600161296838549, 5.797091838674530570792988030, 7.01744807601371376033453947665, 7.86026737991244930719531803925, 9.883822846704352633448833210665, 11.25491669000972543842404174808, 12.25646937156844535097945709716, 13.52099444324612597211000736665, 14.8389546340914285148958638364, 15.87676240738494914000785509526, 16.62090923024176382533142092586, 18.20776718624007391462040528876, 18.447054672104874240273537019265, 20.63360128713030902205644809947, 21.99354691991791607131419565351, 22.61957386670859592505612733227, 23.522697217022547648955030178631, 24.29240410734129635283490683938, 25.74191397266784433499816982990, 26.73644438010649365804329976385, 27.53220107303560781624533469841, 28.97497651474324900364520104584, 30.0765175024440604939062994102, 30.99792089199533857874101832462