L(s) = 1 | + (−0.239 + 0.970i)2-s + (−0.885 − 0.464i)4-s + (0.180 + 0.983i)5-s + (0.998 + 0.0603i)7-s + (0.663 − 0.748i)8-s + (−0.998 − 0.0603i)10-s + (−0.180 + 0.983i)11-s + (−0.885 − 0.464i)13-s + (−0.297 + 0.954i)14-s + (0.568 + 0.822i)16-s + (−0.935 − 0.354i)19-s + (0.297 − 0.954i)20-s + (−0.911 − 0.410i)22-s + (0.707 + 0.707i)23-s + (−0.935 + 0.354i)25-s + (0.663 − 0.748i)26-s + ⋯ |
L(s) = 1 | + (−0.239 + 0.970i)2-s + (−0.885 − 0.464i)4-s + (0.180 + 0.983i)5-s + (0.998 + 0.0603i)7-s + (0.663 − 0.748i)8-s + (−0.998 − 0.0603i)10-s + (−0.180 + 0.983i)11-s + (−0.885 − 0.464i)13-s + (−0.297 + 0.954i)14-s + (0.568 + 0.822i)16-s + (−0.935 − 0.354i)19-s + (0.297 − 0.954i)20-s + (−0.911 − 0.410i)22-s + (0.707 + 0.707i)23-s + (−0.935 + 0.354i)25-s + (0.663 − 0.748i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.938 + 0.345i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.938 + 0.345i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2597908168 + 1.457083046i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2597908168 + 1.457083046i\) |
\(L(1)\) |
\(\approx\) |
\(0.7344828069 + 0.6581431994i\) |
\(L(1)\) |
\(\approx\) |
\(0.7344828069 + 0.6581431994i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 17 | \( 1 \) |
| 79 | \( 1 \) |
good | 2 | \( 1 + (-0.239 + 0.970i)T \) |
| 5 | \( 1 + (0.180 + 0.983i)T \) |
| 7 | \( 1 + (0.998 + 0.0603i)T \) |
| 11 | \( 1 + (-0.180 + 0.983i)T \) |
| 13 | \( 1 + (-0.885 - 0.464i)T \) |
| 19 | \( 1 + (-0.935 - 0.354i)T \) |
| 23 | \( 1 + (0.707 + 0.707i)T \) |
| 29 | \( 1 + (0.787 + 0.616i)T \) |
| 31 | \( 1 + (0.855 - 0.517i)T \) |
| 37 | \( 1 + (0.410 + 0.911i)T \) |
| 41 | \( 1 + (0.983 - 0.180i)T \) |
| 43 | \( 1 + (0.822 + 0.568i)T \) |
| 47 | \( 1 + (0.354 + 0.935i)T \) |
| 53 | \( 1 + (0.663 - 0.748i)T \) |
| 59 | \( 1 + (-0.464 - 0.885i)T \) |
| 61 | \( 1 + (0.911 - 0.410i)T \) |
| 67 | \( 1 + (-0.970 + 0.239i)T \) |
| 71 | \( 1 + (0.998 - 0.0603i)T \) |
| 73 | \( 1 + (0.297 - 0.954i)T \) |
| 83 | \( 1 + (0.464 - 0.885i)T \) |
| 89 | \( 1 + (-0.748 + 0.663i)T \) |
| 97 | \( 1 + (0.410 + 0.911i)T \) |
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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
−18.15085832289638353452041527160, −17.54839036631374192025685996944, −16.845672528575886817762483739508, −16.58121024508826097778300084734, −15.45094802085605731269161190459, −14.47091531774550952409738871981, −13.95526211697680681541297864882, −13.31975689814465980727096923566, −12.41336833806140868418903279690, −12.12105556694469069147692498398, −11.247673509486865289746350411959, −10.66946457477229924609350294708, −9.95207913459940027223918929132, −9.04669255468697451499900986880, −8.55908794575148150718763387784, −8.07852437974061430205730312812, −7.17358589272389864933382324193, −5.88697573500241839839824341039, −5.21804425958131102468042130234, −4.37457729353707783153118195904, −4.12493522496975688811170544458, −2.69618901761837963349336940902, −2.21764130264741833360447193266, −1.20723270715536725361704207905, −0.5554475294323462097418662733,
1.01940647055461130628801579530, 2.103442771931126669195676415252, 2.8080026997994097859914877445, 4.06964520560140509035265029661, 4.776623418601012104451665458474, 5.30986882391879850784580982895, 6.31922768660937536094303373184, 6.86794999452999243166588545508, 7.72860610426442315285049847715, 7.91358636882582503838452365297, 9.063068214430386464704620616507, 9.70443737756230065256223467495, 10.4401012415620894350696270202, 10.95244558956762947727910145208, 11.92133912188833369247670413496, 12.79267009415399244275687490035, 13.5056353236553728648749948116, 14.400321265777515064800302709484, 14.7329603488962413556680356019, 15.25914838347937805787336771847, 15.828094384080551095944370327245, 17.03647749935212808974427419505, 17.57125178706634691650861665622, 17.72002817600664796636161758165, 18.59410118588823230445516513084