L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.5 + 0.866i)4-s + (−0.5 − 0.866i)5-s + (−0.5 + 0.866i)6-s + 7-s + 8-s + (−0.5 + 0.866i)9-s + (−0.5 + 0.866i)10-s + 11-s + 12-s + (−0.5 + 0.866i)13-s + (−0.5 − 0.866i)14-s + (−0.5 + 0.866i)15-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)17-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.5 − 0.866i)3-s + (−0.5 + 0.866i)4-s + (−0.5 − 0.866i)5-s + (−0.5 + 0.866i)6-s + 7-s + 8-s + (−0.5 + 0.866i)9-s + (−0.5 + 0.866i)10-s + 11-s + 12-s + (−0.5 + 0.866i)13-s + (−0.5 − 0.866i)14-s + (−0.5 + 0.866i)15-s + (−0.5 − 0.866i)16-s + (−0.5 − 0.866i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.305 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.305 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2653973789 - 0.3639800608i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2653973789 - 0.3639800608i\) |
\(L(1)\) |
\(\approx\) |
\(0.4992035152 - 0.4022141313i\) |
\(L(1)\) |
\(\approx\) |
\(0.4992035152 - 0.4022141313i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 19 | \( 1 \) |
good | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 3 | \( 1 + (-0.5 - 0.866i)T \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + (-0.5 + 0.866i)T \) |
| 17 | \( 1 + (-0.5 - 0.866i)T \) |
| 23 | \( 1 + (-0.5 + 0.866i)T \) |
| 29 | \( 1 + (-0.5 + 0.866i)T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 + (-0.5 - 0.866i)T \) |
| 43 | \( 1 + (-0.5 - 0.866i)T \) |
| 47 | \( 1 + (-0.5 + 0.866i)T \) |
| 53 | \( 1 + (-0.5 + 0.866i)T \) |
| 59 | \( 1 + (-0.5 - 0.866i)T \) |
| 61 | \( 1 + (-0.5 + 0.866i)T \) |
| 67 | \( 1 + (-0.5 + 0.866i)T \) |
| 71 | \( 1 + (-0.5 - 0.866i)T \) |
| 73 | \( 1 + (-0.5 - 0.866i)T \) |
| 79 | \( 1 + (-0.5 - 0.866i)T \) |
| 83 | \( 1 + T \) |
| 89 | \( 1 + (-0.5 + 0.866i)T \) |
| 97 | \( 1 + (-0.5 - 0.866i)T \) |
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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
−41.1253901054614516133941262833, −39.72713785644655001036063400490, −38.042880905136483901315133727015, −37.23797040928228191804280146842, −35.182945754836544710002587601690, −34.32332760220468950085327732698, −33.353731970792871474749548817303, −32.089972383249371794155173998363, −30.2828022077364862616451162762, −28.17673136074784985593507284243, −27.2216007043267593171072508725, −26.39821209218555236472220813456, −24.577869497894815801446247298544, −23.084835327948264935378218525980, −22.003361277896269211601619608589, −19.86600550830137009502699607557, −18.04739280159696783196787643803, −16.97134418412441583103620362119, −15.255028388258225412799133289879, −14.57003800972870037766640785250, −11.396325399764076250182529561711, −10.10796142252987842080819448827, −8.1872505424540154019873653341, −6.29765489642468131575353719258, −4.40883445095759858523275964106,
1.57320808455217667944576519897, 4.639645630545534199223837216, 7.498422600723132635590660250524, 8.97228608572294362591041132672, 11.428432959283788437662983956906, 12.0908007615320473227409179149, 13.79080829985181288954638451241, 16.6734732907445739545554694964, 17.714719249995459862616200701135, 19.21004735047649087413862767539, 20.35605651057973071520226704261, 22.02787851228323272114484102839, 23.73867106697467176894410671072, 24.93890194818570857552811786818, 27.15357072471301856669666842041, 28.09764486782685251781474558821, 29.31727491172878771802277642022, 30.57284043815652044178886263982, 31.60225228856004011900831156716, 33.93294256575851837093282442897, 35.416076191824706112222999318218, 36.10464510822524108850693114442, 37.379802909321561811105896280132, 39.069391278677512493842139846169, 40.22787418304894314564986293329