L(s) = 1 | + (−0.309 + 0.951i)3-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 + (0.809 − 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 + (0.809 + 0.587i)41-s + 43-s + ⋯ |
L(s) = 1 | + (−0.309 + 0.951i)3-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 + (0.809 − 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 + (0.809 + 0.587i)41-s + 43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.535 + 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.535 + 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.952347149 + 1.073311928i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.952347149 + 1.073311928i\) |
\(L(1)\) |
\(\approx\) |
\(1.061050533 + 0.3390015226i\) |
\(L(1)\) |
\(\approx\) |
\(1.061050533 + 0.3390015226i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (-0.309 + 0.951i)T \) |
| 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
−20.340326812882070473942973130640, −19.67001714194033850284181069436, −18.95104251244879571114460736257, −18.17031466579757621319558105986, −17.51518363139855900783596809916, −16.997426599128607714354041463957, −15.90980388666226697446058417448, −15.249142629956503100141187658441, −14.083185932445922359929006519616, −13.6797188059426338168717661711, −12.749005328449855760703996836, −12.111499001847088224604662871436, −11.31646561680312561666067969272, −10.69660314190389442402614349059, −9.40987389619272907706365139404, −8.84705533256251923859527509581, −7.67552296630533266225568804062, −7.18577885528418384110586558882, −6.32696146027637380491761654368, −5.50074593978163794974947585570, −4.643463826816439554399479691816, −3.380367376750758703136122567698, −2.51674172804563158190065687158, −1.35531216109057041260759396722, −0.695737256655293712972459440083,
0.7231622801997040356808576971, 1.80727784560861708935179370348, 3.32175082162978366225756456814, 3.78774017489333712253903534213, 4.65465571314489837624926493340, 5.83321647094291763049860475252, 6.154943704622629432815459852479, 7.34667725355032576358210958167, 8.624603039404778503585587280150, 8.9450929818184394540214594928, 9.94246509493576664010441872381, 10.75947828805078395747255621584, 11.318286458217387553940395647093, 12.13882797949781786506229893948, 13.033142054109225852727189753722, 14.2192462513067941085019196623, 14.55249918796163017439274706413, 15.47454345072881703865534753845, 16.46402556018242835058753008235, 16.63043320128745782566172280310, 17.533950655557402588026613221966, 18.52641958134875852018434642385, 19.21059876026107898776501676040, 20.1142806217439828585442596957, 20.929115306755887945460103699743