L(s) = 1 | + (0.309 − 0.951i)3-s + 7-s + (−0.809 − 0.587i)9-s + (0.809 − 0.587i)11-s + (0.809 + 0.587i)13-s + (−0.309 − 0.951i)17-s + (−0.309 − 0.951i)19-s + (0.309 − 0.951i)21-s + (−0.809 + 0.587i)23-s + (−0.809 + 0.587i)27-s + (0.309 − 0.951i)29-s + (−0.309 − 0.951i)31-s + (−0.309 − 0.951i)33-s + (0.809 + 0.587i)37-s + (0.809 − 0.587i)39-s + ⋯ |
L(s) = 1 | + (0.309 − 0.951i)3-s + 7-s + (−0.809 − 0.587i)9-s + (0.809 − 0.587i)11-s + (0.809 + 0.587i)13-s + (−0.309 − 0.951i)17-s + (−0.309 − 0.951i)19-s + (0.309 − 0.951i)21-s + (−0.809 + 0.587i)23-s + (−0.809 + 0.587i)27-s + (0.309 − 0.951i)29-s + (−0.309 − 0.951i)31-s + (−0.309 − 0.951i)33-s + (0.809 + 0.587i)37-s + (0.809 − 0.587i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0627 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0627 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.496672563 - 1.405469088i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.496672563 - 1.405469088i\) |
\(L(1)\) |
\(\approx\) |
\(1.232226991 - 0.5520356740i\) |
\(L(1)\) |
\(\approx\) |
\(1.232226991 - 0.5520356740i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (0.309 - 0.951i)T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + (0.809 - 0.587i)T \) |
| 13 | \( 1 + (0.809 + 0.587i)T \) |
| 17 | \( 1 + (-0.309 - 0.951i)T \) |
| 19 | \( 1 + (-0.309 - 0.951i)T \) |
| 23 | \( 1 + (-0.809 + 0.587i)T \) |
| 29 | \( 1 + (0.309 - 0.951i)T \) |
| 31 | \( 1 + (-0.309 - 0.951i)T \) |
| 37 | \( 1 + (0.809 + 0.587i)T \) |
| 41 | \( 1 + (-0.809 - 0.587i)T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 + (0.309 - 0.951i)T \) |
| 53 | \( 1 + (-0.309 + 0.951i)T \) |
| 59 | \( 1 + (0.809 + 0.587i)T \) |
| 61 | \( 1 + (-0.809 + 0.587i)T \) |
| 67 | \( 1 + (0.309 + 0.951i)T \) |
| 71 | \( 1 + (-0.309 + 0.951i)T \) |
| 73 | \( 1 + (0.809 - 0.587i)T \) |
| 79 | \( 1 + (-0.309 + 0.951i)T \) |
| 83 | \( 1 + (0.309 + 0.951i)T \) |
| 89 | \( 1 + (-0.809 + 0.587i)T \) |
| 97 | \( 1 + (-0.309 + 0.951i)T \) |
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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
−30.30862264419532640293262280721, −28.5807105886909724981713833062, −27.68486288873703752784423555483, −27.07949949272555408131758898879, −25.79026336257016687858907915230, −24.99639556023058020011655438709, −23.62546922179372212359129198105, −22.47130613006915796524632183160, −21.479487856027351473103541491920, −20.54295465527892004308478448557, −19.74231505860460231749133842331, −18.13327507520039596381584909196, −17.12351617555178829398989312133, −15.96553654983800279982456743571, −14.790687531104711998382877738095, −14.205143268258181715154736540379, −12.50788632192964946437581504273, −11.11215152589730242123969605364, −10.265951497685155114144105253642, −8.84642071513005989020576287774, −7.98338912289683342198169437500, −6.07939431900958340068805099698, −4.66904180412287706543862698610, −3.64163179465153110049981934980, −1.767833882594145382899210899180,
0.96571343473865083833130206369, 2.336746970799114484203155694647, 4.09931756726872928661928901431, 5.85821055704741978902469288453, 7.05566098311095027172535027458, 8.272876029680603935534272223791, 9.20368625363715182261765811684, 11.273102177754058236377513803833, 11.8064874966569914626161514129, 13.501527673600670961517934864268, 14.037555493140191143877127052708, 15.330394021210494169091770252780, 16.89781334972209884359811100360, 17.91019518041715484898818987909, 18.79221237467406943996833947510, 19.89952965303485321505575495335, 20.89437516621593388075376027965, 22.13694979919809783678840825521, 23.55643445746072216690068575453, 24.223839376274525634776693920155, 25.13572623122384216771573988670, 26.23104516453882658682039714493, 27.39842368966544391737396701680, 28.48977587554544865579579206229, 29.70179068212558991858256055640