L(s) = 1 | + (0.974 − 0.226i)2-s + (0.309 − 0.951i)3-s + (0.897 − 0.441i)4-s + (0.0855 − 0.996i)6-s + (0.774 − 0.633i)8-s + (−0.809 − 0.587i)9-s + (−0.142 − 0.989i)12-s + (0.466 − 0.884i)13-s + (0.610 − 0.791i)16-s + (−0.198 − 0.980i)17-s + (−0.921 − 0.389i)18-s + (0.993 − 0.113i)19-s + (0.959 − 0.281i)23-s + (−0.362 − 0.931i)24-s + (0.254 − 0.967i)26-s + (−0.809 + 0.587i)27-s + ⋯ |
L(s) = 1 | + (0.974 − 0.226i)2-s + (0.309 − 0.951i)3-s + (0.897 − 0.441i)4-s + (0.0855 − 0.996i)6-s + (0.774 − 0.633i)8-s + (−0.809 − 0.587i)9-s + (−0.142 − 0.989i)12-s + (0.466 − 0.884i)13-s + (0.610 − 0.791i)16-s + (−0.198 − 0.980i)17-s + (−0.921 − 0.389i)18-s + (0.993 − 0.113i)19-s + (0.959 − 0.281i)23-s + (−0.362 − 0.931i)24-s + (0.254 − 0.967i)26-s + (−0.809 + 0.587i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.687 - 0.726i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.687 - 0.726i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.678340167 - 3.900623840i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.678340167 - 3.900623840i\) |
\(L(1)\) |
\(\approx\) |
\(1.812089586 - 1.364408369i\) |
\(L(1)\) |
\(\approx\) |
\(1.812089586 - 1.364408369i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.974 - 0.226i)T \) |
| 3 | \( 1 + (0.309 - 0.951i)T \) |
| 13 | \( 1 + (0.466 - 0.884i)T \) |
| 17 | \( 1 + (-0.198 - 0.980i)T \) |
| 19 | \( 1 + (0.993 - 0.113i)T \) |
| 23 | \( 1 + (0.959 - 0.281i)T \) |
| 29 | \( 1 + (0.736 - 0.676i)T \) |
| 31 | \( 1 + (-0.696 + 0.717i)T \) |
| 37 | \( 1 + (0.985 - 0.170i)T \) |
| 41 | \( 1 + (0.516 + 0.856i)T \) |
| 43 | \( 1 + (0.841 + 0.540i)T \) |
| 47 | \( 1 + (-0.921 + 0.389i)T \) |
| 53 | \( 1 + (-0.610 - 0.791i)T \) |
| 59 | \( 1 + (-0.516 + 0.856i)T \) |
| 61 | \( 1 + (0.974 + 0.226i)T \) |
| 67 | \( 1 + (0.654 - 0.755i)T \) |
| 71 | \( 1 + (-0.870 - 0.491i)T \) |
| 73 | \( 1 + (0.564 + 0.825i)T \) |
| 79 | \( 1 + (0.998 + 0.0570i)T \) |
| 83 | \( 1 + (-0.941 + 0.336i)T \) |
| 89 | \( 1 + (-0.415 + 0.909i)T \) |
| 97 | \( 1 + (-0.362 - 0.931i)T \) |
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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
−18.83610339369566305188800683229, −17.66488160040112262470600289879, −16.98341323108723891790957133912, −16.35547700624698806191845454364, −15.83759539706029263879365466872, −15.21337425145938887382941098858, −14.50398886549267352413386598008, −14.07467015475260940546464684335, −13.32537080616335928667463282076, −12.64662012510443749943215043372, −11.72313630649666006174812969514, −11.12194675416383499078636795756, −10.638230516034287028852833418456, −9.65475117362179361389617907079, −8.96635581024810867831949404681, −8.20920857196716096405146704690, −7.43282810559099013020379217770, −6.59002605309186339437386531430, −5.7965729548762261450225068348, −5.17500232481141095561833378649, −4.372318471219815092546471213975, −3.81178212679858683091922249934, −3.14652602749982047695175814401, −2.31177886896822421390704556501, −1.38519023874047244129121314922,
0.81083733944579510053154157268, 1.34000949134938169496908692996, 2.617213912059260184459631076644, 2.85851810737474795884848073748, 3.70581497453080697514076790427, 4.78734654690464180360776910319, 5.42685404973369927775500570213, 6.20873677908100584850313298311, 6.83833826257392363614820818028, 7.56847173530950767611384290451, 8.13320164204559083955158813138, 9.205616510827666340435394014284, 9.86968989380166220611319565534, 11.03403789691454865732085797076, 11.3253090430716606698492073222, 12.182648079088818412830962903388, 12.825955418187720176065293712222, 13.28187828006010782330713246346, 13.96560743168969454002497269453, 14.51472690451850216896361715949, 15.2262980025819198200019393375, 15.97759235287924924496140822990, 16.57597095505540398790959884712, 17.7388049205085866374409305754, 18.097275707127871241591728942867