| L(s) = 1 | + 4-s − 2·7-s + 6·11-s + 4·13-s + 4·16-s − 12·17-s − 4·19-s + 6·23-s − 14·25-s − 2·28-s − 6·29-s − 4·31-s + 14·37-s − 10·43-s + 6·44-s − 24·47-s + 49-s + 4·52-s + 12·53-s + 18·59-s + 20·61-s + 11·64-s + 2·67-s − 12·68-s + 12·71-s + 8·73-s − 4·76-s + ⋯ |
| L(s) = 1 | + 1/2·4-s − 0.755·7-s + 1.80·11-s + 1.10·13-s + 16-s − 2.91·17-s − 0.917·19-s + 1.25·23-s − 2.79·25-s − 0.377·28-s − 1.11·29-s − 0.718·31-s + 2.30·37-s − 1.52·43-s + 0.904·44-s − 3.50·47-s + 1/7·49-s + 0.554·52-s + 1.64·53-s + 2.34·59-s + 2.56·61-s + 11/8·64-s + 0.244·67-s − 1.45·68-s + 1.42·71-s + 0.936·73-s − 0.458·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.132693147\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.132693147\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| good | 2 | $C_2^3$ | \( 1 - T^{2} - 3 T^{4} - p^{2} T^{6} + p^{4} T^{8} \) |
| 5 | $C_2^2$ | \( ( 1 + 7 T^{2} + p^{2} T^{4} )^{2} \) |
| 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 | $D_4\times C_2$ | \( 1 + 12 T + 77 T^{2} + 396 T^{3} + 1752 T^{4} + 396 p T^{5} + 77 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 6 T - 16 T^{2} - 36 T^{3} + 1347 T^{4} - 36 p T^{5} - 16 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 3 T - 20 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + 2 T + 36 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 7 T + 12 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^3$ | \( 1 - 55 T^{2} + 1344 T^{4} - 55 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 10 T + 16 T^{2} - 20 T^{3} + 907 T^{4} - 20 p T^{5} + 16 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 + 12 T + 82 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_{4}$ | \( ( 1 - 6 T + 67 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 18 T + 128 T^{2} - 1404 T^{3} + 15819 T^{4} - 1404 p T^{5} + 128 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 20 T + 205 T^{2} - 1460 T^{3} + 9904 T^{4} - 1460 p T^{5} + 205 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 2 T - 104 T^{2} + 52 T^{3} + 6907 T^{4} + 52 p T^{5} - 104 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 - 6 T - 35 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 - 4 T + 123 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 - 22 T + 252 T^{2} - 22 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 + 6 T + 148 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 12 T - 22 T^{2} + 144 T^{3} + 6819 T^{4} + 144 p T^{5} - 22 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 8 T - 38 T^{2} + 736 T^{3} - 5213 T^{4} + 736 p T^{5} - 38 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.30891599084988899951216624108, −6.88609703555297680844077115328, −6.80515799414133270797056694387, −6.64306575877115645279583854417, −6.49029079214432068725010745250, −6.31367153352645821848031143752, −6.10119069840932667582459461277, −5.89710491337784289447302709654, −5.38688298048962016830168346771, −5.36922599130360424501552094173, −5.03009239526639397128313806933, −4.74337901744352711374104959661, −4.24279220095949821797261467522, −4.22843279020263432811690415438, −3.72700525261561998724332904938, −3.65084687526249175622769557865, −3.60807782800730049140746860940, −3.41776399418811759011358789549, −2.54707369611749812212089553033, −2.45773517442930590152149131707, −2.13368217756690315240089812823, −1.74197926939890262643808917422, −1.71928227525545593051709426587, −0.795358460552956475064080498450, −0.55418257381289744539281443874,
0.55418257381289744539281443874, 0.795358460552956475064080498450, 1.71928227525545593051709426587, 1.74197926939890262643808917422, 2.13368217756690315240089812823, 2.45773517442930590152149131707, 2.54707369611749812212089553033, 3.41776399418811759011358789549, 3.60807782800730049140746860940, 3.65084687526249175622769557865, 3.72700525261561998724332904938, 4.22843279020263432811690415438, 4.24279220095949821797261467522, 4.74337901744352711374104959661, 5.03009239526639397128313806933, 5.36922599130360424501552094173, 5.38688298048962016830168346771, 5.89710491337784289447302709654, 6.10119069840932667582459461277, 6.31367153352645821848031143752, 6.49029079214432068725010745250, 6.64306575877115645279583854417, 6.80515799414133270797056694387, 6.88609703555297680844077115328, 7.30891599084988899951216624108
