L(s) = 1 | + (0.399 − 0.916i)2-s + (−0.809 − 0.587i)3-s + (−0.681 − 0.732i)4-s + (0.958 − 0.285i)5-s + (−0.861 + 0.506i)6-s + (0.215 − 0.976i)7-s + (−0.943 + 0.331i)8-s + (0.309 + 0.951i)9-s + (0.120 − 0.992i)10-s + (0.120 + 0.992i)12-s + (0.309 + 0.951i)13-s + (−0.809 − 0.587i)14-s + (−0.943 − 0.331i)15-s + (−0.0724 + 0.997i)16-s + (−0.168 − 0.985i)17-s + (0.995 + 0.0965i)18-s + ⋯ |
L(s) = 1 | + (0.399 − 0.916i)2-s + (−0.809 − 0.587i)3-s + (−0.681 − 0.732i)4-s + (0.958 − 0.285i)5-s + (−0.861 + 0.506i)6-s + (0.215 − 0.976i)7-s + (−0.943 + 0.331i)8-s + (0.309 + 0.951i)9-s + (0.120 − 0.992i)10-s + (0.120 + 0.992i)12-s + (0.309 + 0.951i)13-s + (−0.809 − 0.587i)14-s + (−0.943 − 0.331i)15-s + (−0.0724 + 0.997i)16-s + (−0.168 − 0.985i)17-s + (0.995 + 0.0965i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.599 - 0.800i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.599 - 0.800i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.419681326 - 0.7100859993i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.419681326 - 0.7100859993i\) |
\(L(1)\) |
\(\approx\) |
\(0.8628285401 - 0.6818175826i\) |
\(L(1)\) |
\(\approx\) |
\(0.8628285401 - 0.6818175826i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 \) |
| 547 | \( 1 \) |
good | 2 | \( 1 + (0.399 - 0.916i)T \) |
| 3 | \( 1 + (-0.809 - 0.587i)T \) |
| 5 | \( 1 + (0.958 - 0.285i)T \) |
| 7 | \( 1 + (0.215 - 0.976i)T \) |
| 13 | \( 1 + (0.309 + 0.951i)T \) |
| 17 | \( 1 + (-0.168 - 0.985i)T \) |
| 19 | \( 1 + (0.644 + 0.764i)T \) |
| 23 | \( 1 + (-0.354 + 0.935i)T \) |
| 29 | \( 1 + (0.836 + 0.548i)T \) |
| 31 | \( 1 + (-0.527 - 0.849i)T \) |
| 37 | \( 1 + (0.836 + 0.548i)T \) |
| 41 | \( 1 + (-0.809 - 0.587i)T \) |
| 43 | \( 1 + (0.120 + 0.992i)T \) |
| 47 | \( 1 + (-0.262 + 0.964i)T \) |
| 53 | \( 1 + (-0.607 + 0.794i)T \) |
| 59 | \( 1 + (0.644 - 0.764i)T \) |
| 61 | \( 1 + (-0.861 + 0.506i)T \) |
| 67 | \( 1 + (0.120 + 0.992i)T \) |
| 71 | \( 1 + (-0.527 + 0.849i)T \) |
| 73 | \( 1 + (0.0241 + 0.999i)T \) |
| 79 | \( 1 + (-0.906 + 0.421i)T \) |
| 83 | \( 1 + (0.958 - 0.285i)T \) |
| 89 | \( 1 + (-0.748 + 0.663i)T \) |
| 97 | \( 1 + (0.715 + 0.698i)T \) |
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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
−17.78048197653893857089075860710, −17.14894198345264951176493983817, −16.43817103336459743337318868211, −15.82062384162029896476775710554, −15.128842411781051682678146431305, −14.85431369986972928556089738431, −14.00413465752281953488153760845, −13.214470520529003390669208701824, −12.6200738744898467950744451457, −12.05463839331037138892035904523, −11.204892552752502606695812145904, −10.41272918499808449414914172614, −9.86942249519060162572446547465, −8.97975008358444566288750385594, −8.60382492299828062815493866019, −7.65679947892547399434980612658, −6.575381262893088687919389205906, −6.28416456588075892245781574477, −5.5820282742302647646672194362, −5.13515052811258091317421207903, −4.472333792220339405687822270002, −3.41433358946158785831895232162, −2.828099967151483682474003988591, −1.736916836922794156851383301625, −0.42817644456649805591676228757,
1.05260522204414275796354929090, 1.29902695285217528218003022636, 2.10211495647235090989372656040, 2.976359648787793660654875967606, 4.04509635563127509034319783134, 4.66395479533391548178015597578, 5.29915582210338477996985871496, 5.99122815873028111233122246249, 6.62010547716928571765666885864, 7.408843631206443854042284398942, 8.273739157381034521185309129716, 9.325715603680274670613359442173, 9.79986044999836150024885839005, 10.40913133148336110212203013665, 11.234171673135354846647369430126, 11.5739967216256279412051077795, 12.31736992647754939229968011218, 13.11560698653860860466298173352, 13.51963411426240285954724600342, 14.07799677254577046793979848220, 14.476285436108123238458694714749, 15.918942067926959329637241072986, 16.397700661064152902068876656604, 17.22046678835180798922504662108, 17.64965744341335305417477648658