| L(s) = 1 | − 2·2-s − 2·3-s + 4-s + 8·5-s + 4·6-s + 2·7-s + 2·8-s + 9-s − 16·10-s − 6·11-s − 2·12-s − 4·14-s − 16·15-s − 4·16-s + 8·17-s − 2·18-s − 6·19-s + 8·20-s − 4·21-s + 12·22-s + 2·23-s − 4·24-s + 26·25-s + 2·27-s + 2·28-s + 2·29-s + 32·30-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 1/2·4-s + 3.57·5-s + 1.63·6-s + 0.755·7-s + 0.707·8-s + 1/3·9-s − 5.05·10-s − 1.80·11-s − 0.577·12-s − 1.06·14-s − 4.13·15-s − 16-s + 1.94·17-s − 0.471·18-s − 1.37·19-s + 1.78·20-s − 0.872·21-s + 2.55·22-s + 0.417·23-s − 0.816·24-s + 26/5·25-s + 0.384·27-s + 0.377·28-s + 0.371·29-s + 5.84·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1744935194\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1744935194\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 13 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( ( 1 - 4 T + 11 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 7 | $D_4\times C_2$ | \( 1 - 2 T - 8 T^{2} + 4 T^{3} + 67 T^{4} + 4 p T^{5} - 8 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 6 T + 8 T^{2} + 36 T^{3} + 267 T^{4} + 36 p T^{5} + 8 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2$$\times$$C_2^2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + 47 T^{2} + 8 p T^{3} + p^{2} T^{4} ) \) |
| 19 | $D_4\times C_2$ | \( 1 + 6 T - 8 T^{2} + 36 T^{3} + 891 T^{4} + 36 p T^{5} - 8 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 2 T - 16 T^{2} + 52 T^{3} - 221 T^{4} + 52 p T^{5} - 16 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 2 T - 43 T^{2} + 22 T^{3} + 1252 T^{4} + 22 p T^{5} - 43 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 + 14 T + 85 T^{2} + 14 p T^{3} + 100 p T^{4} + 14 p^{2} T^{5} + 85 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 + 2 T + 29 T^{2} - 214 T^{3} - 1220 T^{4} - 214 p T^{5} + 29 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 2 T - 8 T^{2} + 148 T^{3} - 1877 T^{4} + 148 p T^{5} - 8 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 + 6 T + 76 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_{4}$ | \( ( 1 + 6 T + 103 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 8 T + 5 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 - 8 T - 47 T^{2} + 88 T^{3} + 4696 T^{4} + 88 p T^{5} - 47 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 2 T + 16 T^{2} - 292 T^{3} - 4613 T^{4} - 292 p T^{5} + 16 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 + 6 T - 112 T^{2} + 36 T^{3} + 14307 T^{4} + 36 p T^{5} - 112 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 16 T + 207 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 + 12 T + 182 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 - 10 T + 164 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 12 T - 58 T^{2} - 288 T^{3} + 20067 T^{4} - 288 p T^{5} - 58 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2^2$ | \( ( 1 - 6 T - 61 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.12237749825226157651942607438, −7.03828032221575775008672475905, −6.53760685400302860304977496968, −6.36687108552210235542888378234, −6.26613876679924249381235977495, −5.84494241582686315342978736907, −5.81282787227736384873495479629, −5.45045609462457575957114398073, −5.38068082764858988781042809711, −5.33992941744246851468231783403, −5.18967739445822727903694396601, −4.67168461357426992200885985600, −4.46290468013006483651713456094, −4.37724996921088537345360347029, −3.77740580666081781485826414909, −3.40568803763984963013532894067, −2.95819985223216164617315160464, −2.85063198901189601756951461318, −2.60764351709029808494873694966, −1.92012898038350777096510378350, −1.84334037570449778381363369033, −1.64468497114695402280470238372, −1.57571721381686050679496760927, −0.950900944682986773641632937442, −0.13119698711258467608477557001,
0.13119698711258467608477557001, 0.950900944682986773641632937442, 1.57571721381686050679496760927, 1.64468497114695402280470238372, 1.84334037570449778381363369033, 1.92012898038350777096510378350, 2.60764351709029808494873694966, 2.85063198901189601756951461318, 2.95819985223216164617315160464, 3.40568803763984963013532894067, 3.77740580666081781485826414909, 4.37724996921088537345360347029, 4.46290468013006483651713456094, 4.67168461357426992200885985600, 5.18967739445822727903694396601, 5.33992941744246851468231783403, 5.38068082764858988781042809711, 5.45045609462457575957114398073, 5.81282787227736384873495479629, 5.84494241582686315342978736907, 6.26613876679924249381235977495, 6.36687108552210235542888378234, 6.53760685400302860304977496968, 7.03828032221575775008672475905, 7.12237749825226157651942607438
