L(s) = 1 | + (−0.990 − 0.139i)2-s + (0.961 + 0.275i)4-s + (−0.173 + 0.984i)5-s + (−0.997 + 0.0697i)7-s + (−0.913 − 0.406i)8-s + (0.309 − 0.951i)10-s + (−0.559 − 0.829i)11-s + (−0.615 − 0.788i)13-s + (0.997 + 0.0697i)14-s + (0.848 + 0.529i)16-s + (−0.913 − 0.406i)17-s + (0.309 − 0.951i)19-s + (−0.438 + 0.898i)20-s + (0.438 + 0.898i)22-s + (−0.559 + 0.829i)23-s + ⋯ |
L(s) = 1 | + (−0.990 − 0.139i)2-s + (0.961 + 0.275i)4-s + (−0.173 + 0.984i)5-s + (−0.997 + 0.0697i)7-s + (−0.913 − 0.406i)8-s + (0.309 − 0.951i)10-s + (−0.559 − 0.829i)11-s + (−0.615 − 0.788i)13-s + (0.997 + 0.0697i)14-s + (0.848 + 0.529i)16-s + (−0.913 − 0.406i)17-s + (0.309 − 0.951i)19-s + (−0.438 + 0.898i)20-s + (0.438 + 0.898i)22-s + (−0.559 + 0.829i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.458 + 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.458 + 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2268122629 + 0.1381958243i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2268122629 + 0.1381958243i\) |
\(L(1)\) |
\(\approx\) |
\(0.4541264413 + 0.01348423063i\) |
\(L(1)\) |
\(\approx\) |
\(0.4541264413 + 0.01348423063i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 31 | \( 1 \) |
good | 2 | \( 1 + (-0.990 - 0.139i)T \) |
| 5 | \( 1 + (-0.173 + 0.984i)T \) |
| 7 | \( 1 + (-0.997 + 0.0697i)T \) |
| 11 | \( 1 + (-0.559 - 0.829i)T \) |
| 13 | \( 1 + (-0.615 - 0.788i)T \) |
| 17 | \( 1 + (-0.913 - 0.406i)T \) |
| 19 | \( 1 + (0.309 - 0.951i)T \) |
| 23 | \( 1 + (-0.559 + 0.829i)T \) |
| 29 | \( 1 + (-0.990 - 0.139i)T \) |
| 37 | \( 1 + (-0.5 - 0.866i)T \) |
| 41 | \( 1 + (-0.0348 - 0.999i)T \) |
| 43 | \( 1 + (0.990 + 0.139i)T \) |
| 47 | \( 1 + (-0.0348 + 0.999i)T \) |
| 53 | \( 1 + (-0.913 - 0.406i)T \) |
| 59 | \( 1 + (0.615 + 0.788i)T \) |
| 61 | \( 1 + (-0.939 - 0.342i)T \) |
| 67 | \( 1 + (-0.939 + 0.342i)T \) |
| 71 | \( 1 + (0.809 - 0.587i)T \) |
| 73 | \( 1 + (0.913 - 0.406i)T \) |
| 79 | \( 1 + (0.961 - 0.275i)T \) |
| 83 | \( 1 + (0.374 + 0.927i)T \) |
| 89 | \( 1 + (0.104 - 0.994i)T \) |
| 97 | \( 1 + (-0.997 + 0.0697i)T \) |
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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
−21.65527354099816117743327863387, −20.546846828393027196156030372140, −20.22964024294154942258743712599, −19.37279306593362228131400888559, −18.68050538091782073729875462922, −17.75746173665736464847452074865, −16.82534874759835038371618359765, −16.431416349530257012812112252332, −15.62251448303691445631539108742, −14.87532359930996142837155491540, −13.60742519324283421434305883164, −12.50635998726956851924280259982, −12.19181731317541303511045526915, −10.98162620243393571044790424857, −9.92095536210029477866411711666, −9.512520471626004141078352164636, −8.597166903680540591338550824672, −7.7524980815283011674903798231, −6.87006765330457247463322577158, −6.0186238295780552737774278608, −4.90031092302284298860229800044, −3.82678964735965908065149833009, −2.42520069307703652575289452728, −1.57852868670644773968988671747, −0.167539393311775800987710758500,
0.45541591941412387514722931722, 2.2742477473588387939176203819, 2.917524095478697224368661618483, 3.70009144148586888471027883162, 5.520976539378821208975624982500, 6.34571076863189356584082233573, 7.25524654854786552086444166314, 7.79154381755307354850355938346, 9.06435757849553054276113308458, 9.656995877553349053333908364534, 10.67644635219392761568200671758, 11.075014166275560520851621676241, 12.11044199032368928395094590418, 13.07398375816981293708487530552, 13.96310566555000237582927728355, 15.32632898638351136524789703175, 15.61672800932672983630355700747, 16.43020416432562643235518627532, 17.61762671559073787242334666995, 18.008215888870671853688694835557, 19.05847074869773917151126682027, 19.422068590161509633524312279302, 20.177296153952308833728827921333, 21.2442057463352658691198478846, 22.20678812249311552809311749975