L(s) = 1 | + (0.755 − 0.654i)3-s + (0.909 + 0.415i)7-s + (0.142 − 0.989i)9-s + (−0.959 − 0.281i)11-s + (0.909 − 0.415i)13-s + (0.540 − 0.841i)17-s + (−0.841 + 0.540i)19-s + (0.959 − 0.281i)21-s + (−0.540 − 0.841i)27-s + (0.841 + 0.540i)29-s + (0.654 − 0.755i)31-s + (−0.909 + 0.415i)33-s + (−0.989 − 0.142i)37-s + (0.415 − 0.909i)39-s + (−0.142 − 0.989i)41-s + ⋯ |
L(s) = 1 | + (0.755 − 0.654i)3-s + (0.909 + 0.415i)7-s + (0.142 − 0.989i)9-s + (−0.959 − 0.281i)11-s + (0.909 − 0.415i)13-s + (0.540 − 0.841i)17-s + (−0.841 + 0.540i)19-s + (0.959 − 0.281i)21-s + (−0.540 − 0.841i)27-s + (0.841 + 0.540i)29-s + (0.654 − 0.755i)31-s + (−0.909 + 0.415i)33-s + (−0.989 − 0.142i)37-s + (0.415 − 0.909i)39-s + (−0.142 − 0.989i)41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.397 - 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.397 - 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.760547710 - 1.155405637i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.760547710 - 1.155405637i\) |
\(L(1)\) |
\(\approx\) |
\(1.407726949 - 0.4392014845i\) |
\(L(1)\) |
\(\approx\) |
\(1.407726949 - 0.4392014845i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 \) |
good | 3 | \( 1 + (0.755 - 0.654i)T \) |
| 7 | \( 1 + (0.909 + 0.415i)T \) |
| 11 | \( 1 + (-0.959 - 0.281i)T \) |
| 13 | \( 1 + (0.909 - 0.415i)T \) |
| 17 | \( 1 + (0.540 - 0.841i)T \) |
| 19 | \( 1 + (-0.841 + 0.540i)T \) |
| 29 | \( 1 + (0.841 + 0.540i)T \) |
| 31 | \( 1 + (0.654 - 0.755i)T \) |
| 37 | \( 1 + (-0.989 - 0.142i)T \) |
| 41 | \( 1 + (-0.142 - 0.989i)T \) |
| 43 | \( 1 + (0.755 - 0.654i)T \) |
| 47 | \( 1 - iT \) |
| 53 | \( 1 + (-0.909 - 0.415i)T \) |
| 59 | \( 1 + (-0.415 - 0.909i)T \) |
| 61 | \( 1 + (0.654 - 0.755i)T \) |
| 67 | \( 1 + (0.281 + 0.959i)T \) |
| 71 | \( 1 + (0.959 - 0.281i)T \) |
| 73 | \( 1 + (0.540 + 0.841i)T \) |
| 79 | \( 1 + (0.415 + 0.909i)T \) |
| 83 | \( 1 + (-0.989 - 0.142i)T \) |
| 89 | \( 1 + (0.654 + 0.755i)T \) |
| 97 | \( 1 + (-0.989 + 0.142i)T \) |
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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
−21.51916539807779204599682663950, −21.26553445928465158969912896881, −20.65712596331812929062131550503, −19.73407973895680275975317182536, −19.023230558128218055452245626, −18.07816503516493648842130066930, −17.252420890230206017559397485772, −16.36503091208609373387302813796, −15.50061973176157259855632608412, −14.95323066602996014719180727242, −14.00792925959583823501236112777, −13.48708086226961523042622222331, −12.482443192026440577994260760317, −11.24275060449484668403500168464, −10.580431056418580240508574084049, −9.983902518376374748232617006835, −8.69528557471761563373404443813, −8.27948544608318444060337179281, −7.44652752430287569426648663305, −6.23824902117532491235231920204, −4.995361970534406898245888065889, −4.41283727396814207785938595298, −3.454064815053214482051619406888, −2.3813008041755093814243508814, −1.40892545930290641662128348720,
0.906039357822287982845232860804, 2.01974252671063654203425216071, 2.83530997336075004973385211333, 3.83416191682369690447612559859, 5.083213598305371720330489672715, 5.92727884932248568074019362718, 6.98404942917805865275200378926, 8.137738714341118559523709119528, 8.21804051395203203680790033096, 9.29468389285825609772590133954, 10.41906433595362656974204071655, 11.24346569083721187914924249818, 12.265454696498479659523824143812, 12.870019404573562125505477412238, 13.94712846453586965769681624020, 14.26045940069563717265930112761, 15.44236485408657041835267595944, 15.83660105664535178842828710851, 17.25268511858847915332155581747, 17.95388914720235925187645616506, 18.7209243252621168625690073073, 19.06704473036263949961370326398, 20.45222035003188159630193813539, 20.78850285472188828997517066985, 21.38154906495619964690869893044