L(s) = 1 | + (0.115 − 0.993i)2-s + (−0.973 − 0.229i)4-s + (−0.876 + 0.482i)5-s + (−0.612 − 0.790i)7-s + (−0.339 + 0.940i)8-s + (0.377 + 0.925i)10-s + (0.0339 − 0.999i)11-s + (−0.999 + 0.0135i)13-s + (−0.855 + 0.517i)14-s + (0.894 + 0.446i)16-s + (0.142 − 0.989i)17-s + (−0.685 + 0.728i)19-s + (0.963 − 0.268i)20-s + (−0.988 − 0.149i)22-s + (0.534 − 0.844i)25-s + (−0.101 + 0.994i)26-s + ⋯ |
L(s) = 1 | + (0.115 − 0.993i)2-s + (−0.973 − 0.229i)4-s + (−0.876 + 0.482i)5-s + (−0.612 − 0.790i)7-s + (−0.339 + 0.940i)8-s + (0.377 + 0.925i)10-s + (0.0339 − 0.999i)11-s + (−0.999 + 0.0135i)13-s + (−0.855 + 0.517i)14-s + (0.894 + 0.446i)16-s + (0.142 − 0.989i)17-s + (−0.685 + 0.728i)19-s + (0.963 − 0.268i)20-s + (−0.988 − 0.149i)22-s + (0.534 − 0.844i)25-s + (−0.101 + 0.994i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.846 - 0.531i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.846 - 0.531i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2305586987 - 0.8006791052i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2305586987 - 0.8006791052i\) |
\(L(1)\) |
\(\approx\) |
\(0.5957711635 - 0.4229760633i\) |
\(L(1)\) |
\(\approx\) |
\(0.5957711635 - 0.4229760633i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 23 | \( 1 \) |
| 29 | \( 1 \) |
good | 2 | \( 1 + (0.115 - 0.993i)T \) |
| 5 | \( 1 + (-0.876 + 0.482i)T \) |
| 7 | \( 1 + (-0.612 - 0.790i)T \) |
| 11 | \( 1 + (0.0339 - 0.999i)T \) |
| 13 | \( 1 + (-0.999 + 0.0135i)T \) |
| 17 | \( 1 + (0.142 - 0.989i)T \) |
| 19 | \( 1 + (-0.685 + 0.728i)T \) |
| 31 | \( 1 + (0.476 - 0.879i)T \) |
| 37 | \( 1 + (0.979 + 0.202i)T \) |
| 41 | \( 1 + (-0.580 + 0.814i)T \) |
| 43 | \( 1 + (0.476 + 0.879i)T \) |
| 47 | \( 1 + (0.733 + 0.680i)T \) |
| 53 | \( 1 + (0.986 + 0.162i)T \) |
| 59 | \( 1 + (0.235 + 0.971i)T \) |
| 61 | \( 1 + (0.195 - 0.980i)T \) |
| 67 | \( 1 + (-0.0339 - 0.999i)T \) |
| 71 | \( 1 + (-0.591 - 0.806i)T \) |
| 73 | \( 1 + (-0.557 - 0.830i)T \) |
| 79 | \( 1 + (0.833 + 0.552i)T \) |
| 83 | \( 1 + (-0.275 + 0.961i)T \) |
| 89 | \( 1 + (0.992 - 0.122i)T \) |
| 97 | \( 1 + (0.546 + 0.837i)T \) |
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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
−17.685404710078866070223118869996, −17.26234457271696836824156598142, −16.62702459302947158731420110552, −15.86426997425836310872248275724, −15.38367890204794708153903837030, −14.9294856621301222149050659304, −14.40049869301336834589165829045, −13.156389189102673016106213426589, −12.859756181170028356800659390561, −12.16213202265474205597818953447, −11.80987355748360898078583053937, −10.44170525662150294605732069231, −9.91218177221389480603010925012, −8.91733594149239608250072512875, −8.75387566488857212263547576164, −7.85765356070888029140221110413, −7.12662839296587555038756694256, −6.73241741539286356758369104813, −5.71605929547211788511317222707, −5.14662592667813124141932136810, −4.386249837030279650774344321964, −3.8872307059781353346368201915, −2.89448180181480840392039570948, −2.007844720270025639854156107240, −0.6394913880285472066242650421,
0.378061332895984212025774294870, 1.02344134400475626709390281804, 2.391113603798904226327061256587, 2.93763275945155339062524475067, 3.58344793323171567204414509837, 4.2672977644992457525722268091, 4.83235548157969142156086672482, 5.947709622374510281545693712089, 6.56854025036435169444554797114, 7.66988324464735551317632466028, 7.90478142304206653754564569199, 8.9634462237938358586621402845, 9.653392867922150065368110195260, 10.28107958952735559230265361593, 10.86982586736484606766339550931, 11.51637510280123341731526525176, 12.04438715021085371873041489823, 12.75130209452927199934223247168, 13.473698033821122079472312915406, 14.047478469951527331035634416223, 14.68590597696605978923085869529, 15.30972796697760896801256053039, 16.424610714307927471390268997870, 16.640847593936124096820440283273, 17.53672738713587703955185055540