L(s) = 1 | + 2-s + 4-s + (0.5 + 0.866i)5-s + 8-s + (0.5 + 0.866i)10-s + (−0.5 + 0.866i)11-s + (0.5 − 0.866i)13-s + 16-s + (0.5 + 0.866i)17-s + (0.5 − 0.866i)19-s + (0.5 + 0.866i)20-s + (−0.5 + 0.866i)22-s + (−0.5 − 0.866i)23-s + (−0.5 + 0.866i)25-s + (0.5 − 0.866i)26-s + ⋯ |
L(s) = 1 | + 2-s + 4-s + (0.5 + 0.866i)5-s + 8-s + (0.5 + 0.866i)10-s + (−0.5 + 0.866i)11-s + (0.5 − 0.866i)13-s + 16-s + (0.5 + 0.866i)17-s + (0.5 − 0.866i)19-s + (0.5 + 0.866i)20-s + (−0.5 + 0.866i)22-s + (−0.5 − 0.866i)23-s + (−0.5 + 0.866i)25-s + (0.5 − 0.866i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.888 + 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.888 + 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(3.059012581 + 0.7418126486i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.059012581 + 0.7418126486i\) |
\(L(1)\) |
\(\approx\) |
\(2.073213055 + 0.2971169318i\) |
\(L(1)\) |
\(\approx\) |
\(2.073213055 + 0.2971169318i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + T \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
| 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 | \( 1 + (-0.5 - 0.866i)T \) |
| 29 | \( 1 + (-0.5 - 0.866i)T \) |
| 31 | \( 1 - T \) |
| 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 | \( 1 - T \) |
| 61 | \( 1 - T \) |
| 67 | \( 1 + T \) |
| 71 | \( 1 + T \) |
| 73 | \( 1 + (0.5 + 0.866i)T \) |
| 79 | \( 1 + T \) |
| 83 | \( 1 + (0.5 + 0.866i)T \) |
| 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
−31.806309246491337715615858272125, −31.321466245309495633999434870677, −29.6403694623016335550090965218, −29.097562584937171783187219128597, −27.85994857972581796900221069019, −26.14602127822975547186726734553, −25.01551314654732998802431370633, −24.114713527671545010090314890406, −23.17201641355782044327165598075, −21.67413628398963260965755127857, −21.01763767270609422819933150069, −19.935347129381804175224038193182, −18.38818487585423253931588011011, −16.553341477517949310499023529821, −16.07170765633500915651228949912, −14.26239526416895353007822646223, −13.476631102819157450096961212610, −12.29147317830405083890155632564, −11.08880352772153036084839562016, −9.43811140145275318813610774010, −7.787923018356822018680152708288, −6.05049240657365185180007576299, −5.06279904105178444604751993433, −3.46881558818835764372243455580, −1.58383548899691464539161993848,
2.13352049364479411492124644125, 3.510890437731257889655225325521, 5.258263908814165500852196519995, 6.477171390374483047820456285708, 7.751033910406363023576183196459, 10.05461166463026957629289341755, 10.99603930091906372719962683054, 12.52758968912776062803079877345, 13.57504247323292285531241344932, 14.78947558159727230259106434498, 15.60705987934156643708050893929, 17.26162938760909661064144142716, 18.51718579991861646061470421524, 20.042499403622263354321625511998, 21.0815095173360558916896934705, 22.24663199866920285897533006434, 22.97556236574444175030583273426, 24.20258662805533466329938492179, 25.548762629155654103453422691880, 26.13939358071978532003162555965, 28.03731381803955155847392930730, 29.13234726686480607195243914184, 30.37123643197133291331888600155, 30.74079694045858754012473744605, 32.33261155476458045267224810913