L(s) = 1 | + (0.559 + 0.829i)2-s + (−0.374 + 0.927i)4-s + (0.766 + 0.642i)5-s + (0.0348 − 0.999i)7-s + (−0.978 + 0.207i)8-s + (−0.104 + 0.994i)10-s + (−0.0348 + 0.999i)11-s + (−0.438 − 0.898i)13-s + (0.848 − 0.529i)14-s + (−0.719 − 0.694i)16-s + (−0.309 + 0.951i)17-s + (0.913 − 0.406i)19-s + (−0.882 + 0.469i)20-s + (−0.848 + 0.529i)22-s + (−0.848 + 0.529i)23-s + ⋯ |
L(s) = 1 | + (0.559 + 0.829i)2-s + (−0.374 + 0.927i)4-s + (0.766 + 0.642i)5-s + (0.0348 − 0.999i)7-s + (−0.978 + 0.207i)8-s + (−0.104 + 0.994i)10-s + (−0.0348 + 0.999i)11-s + (−0.438 − 0.898i)13-s + (0.848 − 0.529i)14-s + (−0.719 − 0.694i)16-s + (−0.309 + 0.951i)17-s + (0.913 − 0.406i)19-s + (−0.882 + 0.469i)20-s + (−0.848 + 0.529i)22-s + (−0.848 + 0.529i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.210 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.210 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.2189088572 + 0.2710740594i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.2189088572 + 0.2710740594i\) |
\(L(1)\) |
\(\approx\) |
\(0.9876518619 + 0.6603654832i\) |
\(L(1)\) |
\(\approx\) |
\(0.9876518619 + 0.6603654832i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 31 | \( 1 \) |
good | 2 | \( 1 + (0.559 + 0.829i)T \) |
| 5 | \( 1 + (0.766 + 0.642i)T \) |
| 7 | \( 1 + (0.0348 - 0.999i)T \) |
| 11 | \( 1 + (-0.0348 + 0.999i)T \) |
| 13 | \( 1 + (-0.438 - 0.898i)T \) |
| 17 | \( 1 + (-0.309 + 0.951i)T \) |
| 19 | \( 1 + (0.913 - 0.406i)T \) |
| 23 | \( 1 + (-0.848 + 0.529i)T \) |
| 29 | \( 1 + (-0.559 - 0.829i)T \) |
| 37 | \( 1 - T \) |
| 41 | \( 1 + (-0.719 + 0.694i)T \) |
| 43 | \( 1 + (0.997 - 0.0697i)T \) |
| 47 | \( 1 + (-0.241 + 0.970i)T \) |
| 53 | \( 1 + (-0.669 - 0.743i)T \) |
| 59 | \( 1 + (0.559 - 0.829i)T \) |
| 61 | \( 1 + (-0.766 + 0.642i)T \) |
| 67 | \( 1 + (0.173 - 0.984i)T \) |
| 71 | \( 1 + (-0.978 + 0.207i)T \) |
| 73 | \( 1 + (-0.309 - 0.951i)T \) |
| 79 | \( 1 + (-0.990 - 0.139i)T \) |
| 83 | \( 1 + (0.997 - 0.0697i)T \) |
| 89 | \( 1 + (-0.669 + 0.743i)T \) |
| 97 | \( 1 + (-0.882 + 0.469i)T \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−21.453475091924973937519511644272, −20.67935683629289530099050142518, −20.03053041358693164969800227652, −18.87609287921491284388322916128, −18.491319141332996609367599326031, −17.58379798210585683107017027338, −16.358564776790773701650819857003, −15.82074353883579675476071575369, −14.5004717276425015273121242345, −13.9765652256103845467590059852, −13.27275913565212192797555128101, −12.16911989203953275482275254265, −11.88462259943222256115802766049, −10.80835500081141453206578762830, −9.759485699519633030264004444773, −9.12805083613168482384610516059, −8.49862263896423283908987885136, −6.83793856838940239722886129640, −5.72643628418813415680411037975, −5.34811954606194545458972158444, −4.33546043394269795672471839910, −3.102601636767707101263147994082, −2.23637875577299305712192896262, −1.37877005095822635861955348465, −0.05659150596336323812824440197,
1.71424550271498886108095904047, 2.92812274093600477613121466680, 3.855617594754853829625394706347, 4.84155565855498091280782898200, 5.74419539635118372904505381336, 6.62385845355253251599439910123, 7.40965973575369403806203231358, 7.98706228975721332695696766648, 9.45131950808934312807357689961, 10.05456127408046932955043803290, 11.017069614096442268157375812444, 12.188901212816627758690988189333, 13.111708268092114893118629368916, 13.64702979740360806280567209526, 14.49851198744121398832682336618, 15.12988712355919292595918624232, 15.94083013745665737606157274815, 17.12199356533700346906653173308, 17.56481765461229991590563897638, 18.0112409380537078043423632132, 19.367283616030337531041323471072, 20.367551270234080388272063636748, 20.98317188732136687272773639320, 22.15916613708944357296001433550, 22.41734226087075471925213226896