L(s) = 1 | + (0.866 + 0.5i)2-s + (0.866 − 0.5i)3-s + (0.5 + 0.866i)4-s + 6-s + i·8-s + (0.5 − 0.866i)9-s + (−0.5 − 0.866i)11-s + (0.866 + 0.5i)12-s + i·13-s + (−0.5 + 0.866i)16-s + (−0.866 + 0.5i)17-s + (0.866 − 0.5i)18-s + (0.5 − 0.866i)19-s − i·22-s + (−0.866 − 0.5i)23-s + (0.5 + 0.866i)24-s + ⋯ |
L(s) = 1 | + (0.866 + 0.5i)2-s + (0.866 − 0.5i)3-s + (0.5 + 0.866i)4-s + 6-s + i·8-s + (0.5 − 0.866i)9-s + (−0.5 − 0.866i)11-s + (0.866 + 0.5i)12-s + i·13-s + (−0.5 + 0.866i)16-s + (−0.866 + 0.5i)17-s + (0.866 − 0.5i)18-s + (0.5 − 0.866i)19-s − i·22-s + (−0.866 − 0.5i)23-s + (0.5 + 0.866i)24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.908 + 0.417i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.908 + 0.417i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.718111101 + 0.5939212068i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.718111101 + 0.5939212068i\) |
\(L(1)\) |
\(\approx\) |
\(2.022687173 + 0.3439784556i\) |
\(L(1)\) |
\(\approx\) |
\(2.022687173 + 0.3439784556i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (0.866 + 0.5i)T \) |
| 3 | \( 1 + (0.866 - 0.5i)T \) |
| 11 | \( 1 + (-0.5 - 0.866i)T \) |
| 13 | \( 1 + iT \) |
| 17 | \( 1 + (-0.866 + 0.5i)T \) |
| 19 | \( 1 + (0.5 - 0.866i)T \) |
| 23 | \( 1 + (-0.866 - 0.5i)T \) |
| 29 | \( 1 - T \) |
| 31 | \( 1 + (-0.5 - 0.866i)T \) |
| 37 | \( 1 + (0.866 + 0.5i)T \) |
| 41 | \( 1 + T \) |
| 43 | \( 1 + iT \) |
| 47 | \( 1 + (0.866 + 0.5i)T \) |
| 53 | \( 1 + (0.866 - 0.5i)T \) |
| 59 | \( 1 + (0.5 + 0.866i)T \) |
| 61 | \( 1 + (-0.5 + 0.866i)T \) |
| 67 | \( 1 + (-0.866 + 0.5i)T \) |
| 71 | \( 1 + T \) |
| 73 | \( 1 + (0.866 - 0.5i)T \) |
| 79 | \( 1 + (0.5 - 0.866i)T \) |
| 83 | \( 1 + iT \) |
| 89 | \( 1 + (0.5 - 0.866i)T \) |
| 97 | \( 1 - iT \) |
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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
−35.89377509879203932391478691188, −33.84273960730530060122891935750, −32.96917005755833550252605032144, −31.80334912458809387038562252908, −31.00485230188198038257288282362, −29.879416518472621948159014955233, −28.40474534221674878824998352695, −27.188152775860030134872747315948, −25.59616376233569921601209715488, −24.567079380133396770061055546504, −22.96795193488978953650872254987, −21.867376268165700036905272143251, −20.51981840154720851774899142914, −19.954979484248753356949264425323, −18.302937061750072741358329909129, −15.93436159450332709011586603405, −15.003608723005802107184334105923, −13.75107163835168794846892456630, −12.54940421952370673502148455321, −10.71454759968517399891237260839, −9.56053378114159156516744300472, −7.53940282731089718088345180747, −5.30810329549512608468832877963, −3.800184639831668899720305117076, −2.25482763699762490408729632759,
2.45955293864876159220403598747, 4.10981455216769926587429169531, 6.195058614860654066255917608380, 7.59617739475376023513821328415, 8.9326723563768611725471447826, 11.40305618408171642726960277184, 12.99469199329751140900192168461, 13.870216007044681308131658372130, 15.09673006095517454118607126277, 16.40798799212429862133596127345, 18.16097632199931193431851527847, 19.6913951972389108501187299340, 20.9722004665645462449841208562, 22.10785158455013824657689698595, 23.96053750711599078301222154631, 24.31453507290520542980915996304, 25.94738826735708365552515402428, 26.53942157582756157533475900835, 28.91438142656687972759492667454, 30.12097472286172158098414098320, 31.14316437994159622782128062278, 31.98429806201945131277114662520, 33.139790922880960360837633984477, 34.5491197196213397853178449292, 35.59910556888870191104038261084