L(s) = 1 | + (−0.929 + 0.369i)5-s + (−0.776 + 0.629i)7-s + (0.387 − 0.922i)11-s + (0.280 − 0.959i)13-s + (0.351 − 0.936i)17-s + (−0.999 − 0.0378i)19-s + (−0.974 − 0.225i)23-s + (0.726 − 0.686i)25-s + (0.974 − 0.225i)29-s + (0.881 + 0.472i)31-s + (0.489 − 0.872i)35-s + (−0.455 + 0.890i)37-s + (0.553 + 0.832i)41-s + (−0.0567 + 0.998i)43-s + (0.954 + 0.298i)47-s + ⋯ |
L(s) = 1 | + (−0.929 + 0.369i)5-s + (−0.776 + 0.629i)7-s + (0.387 − 0.922i)11-s + (0.280 − 0.959i)13-s + (0.351 − 0.936i)17-s + (−0.999 − 0.0378i)19-s + (−0.974 − 0.225i)23-s + (0.726 − 0.686i)25-s + (0.974 − 0.225i)29-s + (0.881 + 0.472i)31-s + (0.489 − 0.872i)35-s + (−0.455 + 0.890i)37-s + (0.553 + 0.832i)41-s + (−0.0567 + 0.998i)43-s + (0.954 + 0.298i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.632 - 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.632 - 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.075814389 - 0.5101434035i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.075814389 - 0.5101434035i\) |
\(L(1)\) |
\(\approx\) |
\(0.8122102720 + 0.01519587542i\) |
\(L(1)\) |
\(\approx\) |
\(0.8122102720 + 0.01519587542i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 167 | \( 1 \) |
good | 5 | \( 1 + (-0.929 + 0.369i)T \) |
| 7 | \( 1 + (-0.776 + 0.629i)T \) |
| 11 | \( 1 + (0.387 - 0.922i)T \) |
| 13 | \( 1 + (0.280 - 0.959i)T \) |
| 17 | \( 1 + (0.351 - 0.936i)T \) |
| 19 | \( 1 + (-0.999 - 0.0378i)T \) |
| 23 | \( 1 + (-0.974 - 0.225i)T \) |
| 29 | \( 1 + (0.974 - 0.225i)T \) |
| 31 | \( 1 + (0.881 + 0.472i)T \) |
| 37 | \( 1 + (-0.455 + 0.890i)T \) |
| 41 | \( 1 + (0.553 + 0.832i)T \) |
| 43 | \( 1 + (-0.0567 + 0.998i)T \) |
| 47 | \( 1 + (0.954 + 0.298i)T \) |
| 53 | \( 1 + (0.914 - 0.404i)T \) |
| 59 | \( 1 + (0.351 + 0.936i)T \) |
| 61 | \( 1 + (-0.988 - 0.150i)T \) |
| 67 | \( 1 + (-0.929 - 0.369i)T \) |
| 71 | \( 1 + (0.243 - 0.969i)T \) |
| 73 | \( 1 + (0.644 + 0.764i)T \) |
| 79 | \( 1 + (0.862 + 0.505i)T \) |
| 83 | \( 1 + (-0.843 + 0.537i)T \) |
| 89 | \( 1 + (0.169 + 0.985i)T \) |
| 97 | \( 1 + (-0.881 + 0.472i)T \) |
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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
−18.62936553843985829652860479760, −17.45756387923816733017349627201, −17.06665142472594492247594832089, −16.34045831447438659633176768394, −15.73446248082792720882924032794, −15.13823276562417774655667721327, −14.295915050172387663193514290, −13.6603272710452051265080303206, −12.74996634699441463568841370361, −12.24153573373950266743952550060, −11.765280661310135618737826911978, −10.63461082167354051770520131488, −10.287186710514225093376504130464, −9.2883254028814919151569739541, −8.71403594932366735753945293850, −7.89120918701380669313934809787, −7.15474206864971364368879567268, −6.58129984819757274909882839468, −5.80389767175531620972311488186, −4.58415278334904028153227308954, −4.0145743503664121230390693831, −3.71753898833704825685714653086, −2.41539700034168479721349470178, −1.54326836762218776534144354074, −0.55379371573022806274038676204,
0.3288906288850655008985928144, 1.08023022749850493184645076079, 2.70024007422184071448212124867, 2.91655137590314534489280618275, 3.80586283387791508356265063625, 4.57954728175625804799102482955, 5.60291466049978096165925087418, 6.32841513745225195563988285040, 6.80331872900211152433195120832, 7.98675764155122448864599387677, 8.28852667541624345393192535420, 9.0961654943666258244702919989, 10.01986105883714575165508111931, 10.61873640913348638901286474161, 11.41355409348694860038553644025, 12.11055067083328121242095907472, 12.50123045949872260061777795795, 13.53691262789884220949408085636, 14.05060388926240349101769815490, 15.09748662790634932007211023587, 15.39840043644791204050691109324, 16.28673789449219934293974764310, 16.48221880623174731935996699276, 17.70915610688025668625095698731, 18.31782965489067067978077374738