L(s) = 1 | + (−0.293 − 0.955i)2-s + (−0.309 + 0.951i)3-s + (−0.827 + 0.561i)4-s + (−0.0563 + 0.998i)5-s + (0.999 + 0.0161i)6-s + (0.0884 − 0.996i)7-s + (0.779 + 0.626i)8-s + (−0.809 − 0.587i)9-s + (0.970 − 0.239i)10-s + (−0.278 − 0.960i)12-s + (−0.913 + 0.406i)13-s + (−0.978 + 0.207i)14-s + (−0.932 − 0.362i)15-s + (0.369 − 0.929i)16-s + (0.759 − 0.650i)17-s + (−0.324 + 0.945i)18-s + ⋯ |
L(s) = 1 | + (−0.293 − 0.955i)2-s + (−0.309 + 0.951i)3-s + (−0.827 + 0.561i)4-s + (−0.0563 + 0.998i)5-s + (0.999 + 0.0161i)6-s + (0.0884 − 0.996i)7-s + (0.779 + 0.626i)8-s + (−0.809 − 0.587i)9-s + (0.970 − 0.239i)10-s + (−0.278 − 0.960i)12-s + (−0.913 + 0.406i)13-s + (−0.978 + 0.207i)14-s + (−0.932 − 0.362i)15-s + (0.369 − 0.929i)16-s + (0.759 − 0.650i)17-s + (−0.324 + 0.945i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.625 - 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.625 - 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1735011714 - 0.3615017003i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1735011714 - 0.3615017003i\) |
\(L(1)\) |
\(\approx\) |
\(0.6484154598 - 0.06907318621i\) |
\(L(1)\) |
\(\approx\) |
\(0.6484154598 - 0.06907318621i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 \) |
| 547 | \( 1 \) |
good | 2 | \( 1 + (-0.293 - 0.955i)T \) |
| 3 | \( 1 + (-0.309 + 0.951i)T \) |
| 5 | \( 1 + (-0.0563 + 0.998i)T \) |
| 7 | \( 1 + (0.0884 - 0.996i)T \) |
| 13 | \( 1 + (-0.913 + 0.406i)T \) |
| 17 | \( 1 + (0.759 - 0.650i)T \) |
| 19 | \( 1 + (0.769 - 0.638i)T \) |
| 23 | \( 1 + (-0.948 - 0.316i)T \) |
| 29 | \( 1 + (-0.399 + 0.916i)T \) |
| 31 | \( 1 + (0.443 + 0.896i)T \) |
| 37 | \( 1 + (-0.594 - 0.804i)T \) |
| 41 | \( 1 + (0.669 + 0.743i)T \) |
| 43 | \( 1 + (0.692 - 0.721i)T \) |
| 47 | \( 1 + (0.870 + 0.493i)T \) |
| 53 | \( 1 + (-0.704 - 0.709i)T \) |
| 59 | \( 1 + (0.769 + 0.638i)T \) |
| 61 | \( 1 + (-0.999 - 0.0161i)T \) |
| 67 | \( 1 + (0.692 - 0.721i)T \) |
| 71 | \( 1 + (0.554 + 0.832i)T \) |
| 73 | \( 1 + (-0.541 + 0.840i)T \) |
| 79 | \( 1 + (0.644 + 0.764i)T \) |
| 83 | \( 1 + (-0.892 + 0.450i)T \) |
| 89 | \( 1 + (-0.120 - 0.992i)T \) |
| 97 | \( 1 + (-0.877 + 0.478i)T \) |
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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
−17.77563416177256530647811060816, −17.21658796138703462479364990877, −16.85922287116150597702756295565, −16.000090013574145990154731953892, −15.50512895254433940655080282878, −14.68661472932289885547484417719, −14.06564314185092954502383691355, −13.35712030153285479954838634494, −12.6846099034034734367624652481, −12.11923871571907024675227972942, −11.73756477119863695662485172659, −10.498366088966576392333371425868, −9.67744173101792800119436786217, −9.19383336356926225927034616510, −8.25195560336848345905426804694, −7.88371366799180926695298327366, −7.485332924283552531886374168619, −6.26416356438103935988812416372, −5.80529225389108007810097873529, −5.38677639350559423778956146897, −4.65934867894831709297958219805, −3.66787946937607499503193206589, −2.41223805217816546826299513741, −1.6600720640967479685410009422, −0.87241662788690263796478434435,
0.15358698071657867161827611209, 1.17483181615757459560174084215, 2.37093167010821112373261410456, 3.01536740571979831570304613512, 3.644378240441440078531099232484, 4.28977943251416898466910085561, 4.97378848890963758373429904230, 5.72295096189226554311125500773, 6.97015051672881472455053618685, 7.32940298432966709967982774595, 8.20753696892555060167854244244, 9.18786042450069565238452068651, 9.820783474356646404692858204624, 10.129519116978694034401308405, 10.9272941937027606918273843756, 11.22473483534496284460276917156, 12.02766630982484084707103649311, 12.56075267803703468296931634267, 13.86620782180910198481072499737, 14.13457626441478348157710390677, 14.55225947935381851150052613910, 15.66860702667414883296138695394, 16.28880760718882958587830279410, 16.91250404775911231250437966677, 17.66233307727702513597772219599