L(s) = 1 | + (−0.168 − 0.985i)2-s + (0.926 + 0.377i)3-s + (−0.943 + 0.331i)4-s + (−0.0724 + 0.997i)5-s + (0.215 − 0.976i)6-s + (−0.861 + 0.506i)7-s + (0.485 + 0.873i)8-s + (0.715 + 0.698i)9-s + (0.995 − 0.0965i)10-s + (−0.998 + 0.0483i)11-s + (−0.998 − 0.0483i)12-s + (0.995 + 0.0965i)13-s + (0.644 + 0.764i)14-s + (−0.443 + 0.896i)15-s + (0.779 − 0.626i)16-s + (0.836 + 0.548i)17-s + ⋯ |
L(s) = 1 | + (−0.168 − 0.985i)2-s + (0.926 + 0.377i)3-s + (−0.943 + 0.331i)4-s + (−0.0724 + 0.997i)5-s + (0.215 − 0.976i)6-s + (−0.861 + 0.506i)7-s + (0.485 + 0.873i)8-s + (0.715 + 0.698i)9-s + (0.995 − 0.0965i)10-s + (−0.998 + 0.0483i)11-s + (−0.998 − 0.0483i)12-s + (0.995 + 0.0965i)13-s + (0.644 + 0.764i)14-s + (−0.443 + 0.896i)15-s + (0.779 − 0.626i)16-s + (0.836 + 0.548i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 131 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.898 + 0.439i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 131 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.898 + 0.439i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.034219650 + 0.2394209242i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.034219650 + 0.2394209242i\) |
\(L(1)\) |
\(\approx\) |
\(1.060145920 + 0.0004807760724i\) |
\(L(1)\) |
\(\approx\) |
\(1.060145920 + 0.0004807760724i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 131 | \( 1 \) |
good | 2 | \( 1 + (-0.168 - 0.985i)T \) |
| 3 | \( 1 + (0.926 + 0.377i)T \) |
| 5 | \( 1 + (-0.0724 + 0.997i)T \) |
| 7 | \( 1 + (-0.861 + 0.506i)T \) |
| 11 | \( 1 + (-0.998 + 0.0483i)T \) |
| 13 | \( 1 + (0.995 + 0.0965i)T \) |
| 17 | \( 1 + (0.836 + 0.548i)T \) |
| 19 | \( 1 + (-0.354 + 0.935i)T \) |
| 23 | \( 1 + (-0.681 - 0.732i)T \) |
| 29 | \( 1 + (0.715 - 0.698i)T \) |
| 31 | \( 1 + (0.981 - 0.192i)T \) |
| 37 | \( 1 + (-0.607 - 0.794i)T \) |
| 41 | \( 1 + (0.779 + 0.626i)T \) |
| 43 | \( 1 + (-0.527 - 0.849i)T \) |
| 47 | \( 1 + (0.885 - 0.464i)T \) |
| 53 | \( 1 + (0.309 - 0.951i)T \) |
| 59 | \( 1 + (-0.262 + 0.964i)T \) |
| 61 | \( 1 + (-0.809 + 0.587i)T \) |
| 67 | \( 1 + (-0.527 + 0.849i)T \) |
| 71 | \( 1 + (0.120 - 0.992i)T \) |
| 73 | \( 1 + (-0.809 - 0.587i)T \) |
| 79 | \( 1 + (-0.970 + 0.239i)T \) |
| 83 | \( 1 + (0.958 - 0.285i)T \) |
| 89 | \( 1 + (0.309 + 0.951i)T \) |
| 97 | \( 1 + (0.399 - 0.916i)T \) |
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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
−28.43492599893481764101297690839, −27.42965098032201940520964401948, −26.13299512749336229426959711634, −25.7442278257129860285640081488, −24.76803538648357167826631953723, −23.6719640632281962550313062506, −23.2517446449759526108423262798, −21.46177130059565141213115853969, −20.36709661291322916398416243651, −19.408525737029945782651100254010, −18.44248815805824769750784712475, −17.30376476861604078861846951090, −15.904128184194948810810153578446, −15.70311108589455752842304350923, −13.91972013517293116740681700245, −13.37908507813202719890974602644, −12.494653804689745602770896184, −10.19957043436310813027594556044, −9.19490615441808783205810435670, −8.26819034407800645243868207749, −7.36259893600202319927514387692, −6.070766857421256388260489418945, −4.61918035135654122870386062100, −3.2783779724224561312750676583, −1.00652771392818414722411978895,
2.18851987689614023478865865651, 3.11721457178432953990150350227, 4.02431361902891891399867011429, 5.950816229442977197295649712502, 7.78970835646258841906818075773, 8.70742722208499121886964083813, 10.19482711141102084971268454902, 10.3747502792924435417884031777, 12.06394768545018766849625878806, 13.21236905659272838908622556577, 14.11409126930906334840558465696, 15.24894014711020827221313304246, 16.32627554412748730771995983490, 18.180331570775163911881619913431, 18.85247216357514561397268731037, 19.486052331965765410438799562517, 20.8562240298319149606813225529, 21.39929185867863556924271676755, 22.53883279648350205341428525890, 23.33085323984731206063212727653, 25.302660436070756753592619908023, 26.08387686286558313360253283979, 26.600386502881581164956918281519, 27.841266890959620743933620867392, 28.71756842313094289064646401538