L(s) = 1 | + 2-s + (0.5 + 0.866i)3-s + 4-s + (−0.5 + 0.866i)5-s + (0.5 + 0.866i)6-s + (−0.5 − 0.866i)7-s + 8-s + (−0.5 + 0.866i)9-s + (−0.5 + 0.866i)10-s + (0.5 − 0.866i)11-s + (0.5 + 0.866i)12-s + (0.5 − 0.866i)13-s + (−0.5 − 0.866i)14-s − 15-s + 16-s + (0.5 + 0.866i)17-s + ⋯ |
L(s) = 1 | + 2-s + (0.5 + 0.866i)3-s + 4-s + (−0.5 + 0.866i)5-s + (0.5 + 0.866i)6-s + (−0.5 − 0.866i)7-s + 8-s + (−0.5 + 0.866i)9-s + (−0.5 + 0.866i)10-s + (0.5 − 0.866i)11-s + (0.5 + 0.866i)12-s + (0.5 − 0.866i)13-s + (−0.5 − 0.866i)14-s − 15-s + 16-s + (0.5 + 0.866i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.695 + 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.695 + 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.371469482 + 1.005786976i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.371469482 + 1.005786976i\) |
\(L(1)\) |
\(\approx\) |
\(1.879213610 + 0.5376320614i\) |
\(L(1)\) |
\(\approx\) |
\(1.879213610 + 0.5376320614i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 31 | \( 1 \) |
good | 2 | \( 1 + T \) |
| 3 | \( 1 + (0.5 + 0.866i)T \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 + (-0.5 - 0.866i)T \) |
| 11 | \( 1 + (0.5 - 0.866i)T \) |
| 13 | \( 1 + (0.5 - 0.866i)T \) |
| 17 | \( 1 + (0.5 + 0.866i)T \) |
| 19 | \( 1 + (-0.5 - 0.866i)T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| 37 | \( 1 + (0.5 + 0.866i)T \) |
| 41 | \( 1 + (-0.5 + 0.866i)T \) |
| 43 | \( 1 + (0.5 + 0.866i)T \) |
| 47 | \( 1 + T \) |
| 53 | \( 1 + (0.5 - 0.866i)T \) |
| 59 | \( 1 + (-0.5 - 0.866i)T \) |
| 61 | \( 1 - T \) |
| 67 | \( 1 + (-0.5 + 0.866i)T \) |
| 71 | \( 1 + (-0.5 + 0.866i)T \) |
| 73 | \( 1 + (0.5 - 0.866i)T \) |
| 79 | \( 1 + (0.5 + 0.866i)T \) |
| 83 | \( 1 + (0.5 - 0.866i)T \) |
| 89 | \( 1 - T \) |
| 97 | \( 1 + T \) |
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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
−36.03694199047081809395459724281, −35.30864266593074978771591649753, −33.8241549331911259626327357322, −32.129778633924270460121683213657, −31.54489954313244131089796391377, −30.53263976307618275070690829051, −29.16141499249758529834513709123, −28.0945904790797707352893286350, −25.69070979248761182016100995745, −24.904979113067120430837549644393, −23.806582542939423420906081340444, −22.72252967739608720611109521201, −20.96390856165423950362814455136, −19.94453395593242918465398300995, −18.735126484161719053692078780397, −16.60173698608234703190103783223, −15.25089922348459432110354945030, −13.879620718504668161994951672300, −12.44207778956765682717281271220, −11.949121923483919535504377485389, −9.11916197987811066740563336970, −7.45087480449111406298688286638, −5.89688076625617776774994700343, −3.90954114053089609500609102833, −1.93601721019922475920401420063,
3.14900146449111918990716082501, 4.02909061054478038079593727430, 6.16852612519315094017700013557, 7.90843428682917089599907565647, 10.30301459129209214350523835223, 11.220089817316603931959723291699, 13.32190208552506637014404100721, 14.4786587114416088206168908723, 15.53818190195904911978227263916, 16.717895780434211211143117394807, 19.33388309535322849716732648173, 20.20684450635459460249475492654, 21.71749720389530663485905264589, 22.57729055762610550776387144847, 23.77646257686247141114177455735, 25.59696995015023513032903862194, 26.44017494393920563813497510448, 27.88533073949353888140756660178, 29.83973207742039441096728534060, 30.51459276150292375835453676759, 32.05994899404862366440308541101, 32.667001701834777095473969683222, 33.91330850741717403494440774979, 35.06770270199252064084748377345, 37.1672086619915527437976882128