L(s) = 1 | + (0.370 + 0.928i)2-s + (−0.994 + 0.108i)3-s + (−0.725 + 0.687i)4-s + (0.647 + 0.762i)5-s + (−0.468 − 0.883i)6-s + (−0.856 + 0.515i)7-s + (−0.907 − 0.419i)8-s + (0.976 − 0.214i)9-s + (−0.468 + 0.883i)10-s + (−0.0541 − 0.998i)11-s + (0.647 − 0.762i)12-s + (−0.976 − 0.214i)13-s + (−0.796 − 0.605i)14-s + (−0.725 − 0.687i)15-s + (0.0541 − 0.998i)16-s + (−0.856 − 0.515i)17-s + ⋯ |
L(s) = 1 | + (0.370 + 0.928i)2-s + (−0.994 + 0.108i)3-s + (−0.725 + 0.687i)4-s + (0.647 + 0.762i)5-s + (−0.468 − 0.883i)6-s + (−0.856 + 0.515i)7-s + (−0.907 − 0.419i)8-s + (0.976 − 0.214i)9-s + (−0.468 + 0.883i)10-s + (−0.0541 − 0.998i)11-s + (0.647 − 0.762i)12-s + (−0.976 − 0.214i)13-s + (−0.796 − 0.605i)14-s + (−0.725 − 0.687i)15-s + (0.0541 − 0.998i)16-s + (−0.856 − 0.515i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.2166425081 + 0.4184253402i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.2166425081 + 0.4184253402i\) |
\(L(1)\) |
\(\approx\) |
\(0.4818734731 + 0.4923280208i\) |
\(L(1)\) |
\(\approx\) |
\(0.4818734731 + 0.4923280208i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 59 | \( 1 \) |
good | 2 | \( 1 + (0.370 + 0.928i)T \) |
| 3 | \( 1 + (-0.994 + 0.108i)T \) |
| 5 | \( 1 + (0.647 + 0.762i)T \) |
| 7 | \( 1 + (-0.856 + 0.515i)T \) |
| 11 | \( 1 + (-0.0541 - 0.998i)T \) |
| 13 | \( 1 + (-0.976 - 0.214i)T \) |
| 17 | \( 1 + (-0.856 - 0.515i)T \) |
| 19 | \( 1 + (0.267 + 0.963i)T \) |
| 23 | \( 1 + (0.561 - 0.827i)T \) |
| 29 | \( 1 + (-0.370 + 0.928i)T \) |
| 31 | \( 1 + (-0.267 + 0.963i)T \) |
| 37 | \( 1 + (-0.907 + 0.419i)T \) |
| 41 | \( 1 + (-0.561 - 0.827i)T \) |
| 43 | \( 1 + (-0.0541 + 0.998i)T \) |
| 47 | \( 1 + (-0.647 + 0.762i)T \) |
| 53 | \( 1 + (0.468 + 0.883i)T \) |
| 61 | \( 1 + (0.370 + 0.928i)T \) |
| 67 | \( 1 + (-0.907 - 0.419i)T \) |
| 71 | \( 1 + (0.647 - 0.762i)T \) |
| 73 | \( 1 + (-0.796 - 0.605i)T \) |
| 79 | \( 1 + (-0.994 - 0.108i)T \) |
| 83 | \( 1 + (0.947 + 0.319i)T \) |
| 89 | \( 1 + (0.370 - 0.928i)T \) |
| 97 | \( 1 + (-0.796 + 0.605i)T \) |
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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
−31.76602525320418334690073165802, −30.29052895704955771075172503349, −29.26769150538593120765719696677, −28.68621980528588424532368194261, −27.82371742613712322120976142745, −26.34873851876043757273644669564, −24.56659050099921990559114521668, −23.55233622489913863089342286466, −22.45596336479753775292758737950, −21.66675158851064606673130882244, −20.33052022818261517262790897826, −19.308908951235839012320122115, −17.704957737895712717883258862285, −17.03718102850737879498118850499, −15.36587641665881325709006591558, −13.384731365110839176398526934069, −12.8157615573663753901389257998, −11.6309088861351712513465454599, −10.149048044726346741781938021473, −9.42098804891719210934385269104, −6.85245958572027679313066941645, −5.35916828263924095209342322224, −4.2972696366607554351373096163, −1.98089647941108577453101869444, −0.24627041278601672726660186834,
3.158608316527950843814506542938, 5.15758690611459908356299898312, 6.17540475379764428440319769182, 7.07279220095090582428329536189, 9.14269363565060747018718106497, 10.46615055793301507081530368714, 12.12588000172841219934242642384, 13.27144495520562871638744827077, 14.633929538138608299734213053258, 15.91610537074739528940503588002, 16.813274570916788401096276773033, 18.0118330997216399131048477581, 18.87149727049699928977138782977, 21.356913498129085566175122086389, 22.29442940259338328350512761078, 22.68888807604500437814004847637, 24.23705030187170551055270621401, 25.12464085942410437562220164213, 26.48383679389417808931784898634, 27.24267433242063212528051317154, 29.02334572182275404194511683540, 29.60093169943891749351881759061, 31.18622490622001532210019191223, 32.421627345065258890558296057475, 33.257626796419570256633349779685