L(s) = 1 | + (−0.681 − 0.732i)2-s + (−0.861 + 0.506i)3-s + (−0.0724 + 0.997i)4-s + (0.995 + 0.0965i)5-s + (0.958 + 0.285i)6-s + (−0.943 − 0.331i)7-s + (0.779 − 0.626i)8-s + (0.485 − 0.873i)9-s + (−0.607 − 0.794i)10-s + (−0.443 + 0.896i)11-s + (−0.443 − 0.896i)12-s + (−0.607 + 0.794i)13-s + (0.399 + 0.916i)14-s + (−0.906 + 0.421i)15-s + (−0.989 − 0.144i)16-s + (0.715 + 0.698i)17-s + ⋯ |
L(s) = 1 | + (−0.681 − 0.732i)2-s + (−0.861 + 0.506i)3-s + (−0.0724 + 0.997i)4-s + (0.995 + 0.0965i)5-s + (0.958 + 0.285i)6-s + (−0.943 − 0.331i)7-s + (0.779 − 0.626i)8-s + (0.485 − 0.873i)9-s + (−0.607 − 0.794i)10-s + (−0.443 + 0.896i)11-s + (−0.443 − 0.896i)12-s + (−0.607 + 0.794i)13-s + (0.399 + 0.916i)14-s + (−0.906 + 0.421i)15-s + (−0.989 − 0.144i)16-s + (0.715 + 0.698i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 131 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.584 + 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 131 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.584 + 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4647473384 + 0.2379046144i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4647473384 + 0.2379046144i\) |
\(L(1)\) |
\(\approx\) |
\(0.5923243781 + 0.04583792132i\) |
\(L(1)\) |
\(\approx\) |
\(0.5923243781 + 0.04583792132i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 131 | \( 1 \) |
good | 2 | \( 1 + (-0.681 - 0.732i)T \) |
| 3 | \( 1 + (-0.861 + 0.506i)T \) |
| 5 | \( 1 + (0.995 + 0.0965i)T \) |
| 7 | \( 1 + (-0.943 - 0.331i)T \) |
| 11 | \( 1 + (-0.443 + 0.896i)T \) |
| 13 | \( 1 + (-0.607 + 0.794i)T \) |
| 17 | \( 1 + (0.715 + 0.698i)T \) |
| 19 | \( 1 + (0.885 + 0.464i)T \) |
| 23 | \( 1 + (-0.998 - 0.0483i)T \) |
| 29 | \( 1 + (0.485 + 0.873i)T \) |
| 31 | \( 1 + (-0.262 + 0.964i)T \) |
| 37 | \( 1 + (0.644 - 0.764i)T \) |
| 41 | \( 1 + (-0.989 + 0.144i)T \) |
| 43 | \( 1 + (0.215 + 0.976i)T \) |
| 47 | \( 1 + (0.120 + 0.992i)T \) |
| 53 | \( 1 + (-0.809 + 0.587i)T \) |
| 59 | \( 1 + (-0.168 - 0.985i)T \) |
| 61 | \( 1 + (0.309 + 0.951i)T \) |
| 67 | \( 1 + (0.215 - 0.976i)T \) |
| 71 | \( 1 + (-0.354 - 0.935i)T \) |
| 73 | \( 1 + (0.309 - 0.951i)T \) |
| 79 | \( 1 + (-0.748 - 0.663i)T \) |
| 83 | \( 1 + (0.926 - 0.377i)T \) |
| 89 | \( 1 + (-0.809 - 0.587i)T \) |
| 97 | \( 1 + (0.0241 - 0.999i)T \) |
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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
−28.69276669295536515469517907634, −27.61219400329542144414946381497, −26.44866292192254105468696278859, −25.36481996649456398801564571146, −24.71551652789097809281001477141, −23.799611467006817449190385296264, −22.56867469641745703227182221593, −21.92077073472778249921948201209, −20.205433822735411704502090790389, −18.87102427428448918032094220050, −18.29645587275985192425653469882, −17.27376626309884179308454325953, −16.461387016020411167317416208372, −15.59942107191430939233874193597, −13.90261117359283997730381932234, −13.15815803347869285101855990390, −11.72835178376023919584924343620, −10.24172278735823407401734341356, −9.669545097406412038278626897497, −8.10864623285099137317420584309, −6.88734176305079714360920223146, −5.805828361483398917340888549586, −5.314499102812151603681368317892, −2.50722585806510530648074542859, −0.68080246206717958110423274567,
1.60004602982039797075322738616, 3.252438730757434134895301337187, 4.69632211964589971423502589856, 6.210133099178459630587596964429, 7.35432952537490180900331818953, 9.34258941715471162069457878037, 9.93899981331812395329631230232, 10.612014609985492440959399528221, 12.15125510752471375702238565642, 12.77459995586061656776482551558, 14.25385989082099585262535585810, 16.05058752714175782279490619168, 16.73938404547115627376531022054, 17.6928758534368842111519531292, 18.45743886471680947214233760902, 19.80212816614525343853671466091, 20.8705961250210921816804886368, 21.775931505454829265525835793017, 22.42564089020111040765161190829, 23.57485772806566794969854806066, 25.28806079925163250789563357923, 26.12498031076121444357092600284, 26.86010801852171539341827578525, 28.20496087191090938105167826490, 28.80166084391936857587696374011