| L(s) = 1 | − 2·2-s + 2·4-s − 4·8-s − 6·9-s + 8·16-s + 12·18-s + 24·19-s − 2·25-s + 4·29-s − 24·31-s − 8·32-s − 12·36-s − 24·37-s − 48·38-s − 2·49-s + 4·50-s + 8·53-s − 8·58-s + 48·62-s + 8·64-s + 24·72-s + 48·74-s + 48·76-s + 9·81-s + 48·83-s + 4·98-s − 4·100-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s − 1.41·8-s − 2·9-s + 2·16-s + 2.82·18-s + 5.50·19-s − 2/5·25-s + 0.742·29-s − 4.31·31-s − 1.41·32-s − 2·36-s − 3.94·37-s − 7.78·38-s − 2/7·49-s + 0.565·50-s + 1.09·53-s − 1.05·58-s + 6.09·62-s + 64-s + 2.82·72-s + 5.57·74-s + 5.50·76-s + 81-s + 5.26·83-s + 0.404·98-s − 2/5·100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3905249022\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3905249022\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| good | 3 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 30 T^{2} + 419 T^{4} - 30 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} )( 1 + 4 T + 3 T^{2} + 4 p T^{3} + p^{2} T^{4} ) \) |
| 17 | $D_4\times C_2$ | \( 1 - 26 T^{2} + 315 T^{4} - 26 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 23 | $D_4\times C_2$ | \( 1 - 60 T^{2} + 1766 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 2 T + 11 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 37 | $D_{4}$ | \( ( 1 + 12 T + 98 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 70 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 91 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 59 | $C_2^2$ | \( ( 1 + 106 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 - 148 T^{2} + 11190 T^{4} - 148 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2}( 1 + 16 T + p T^{2} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 228 T^{2} + 22310 T^{4} - 228 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 124 T^{2} + 7590 T^{4} - 124 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 94 T^{2} + 3891 T^{4} - 94 p^{2} T^{6} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 24 T + 298 T^{2} - 24 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 260 T^{2} + 31014 T^{4} - 260 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 154 T^{2} + 20859 T^{4} - 154 p^{2} T^{6} + p^{4} T^{8} \) |
| 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
−9.559794286102011012517227700707, −9.404360551564499699705224351273, −9.002827187948285875855316348389, −8.969596023014252945070630651690, −8.868877961939504159867187409771, −8.228199031050160078963874641245, −8.009373213722982671850439819836, −8.000783233667318446666613834148, −7.34476711637577239965022710427, −7.14203736574820061046078816135, −7.11378647964160262014460832949, −6.81842044350380925908750047663, −6.02985952146808533006216944611, −5.65956700888107471052541921644, −5.54146860361022471232453573477, −5.51683083249260160026979033807, −5.16220767501999316703122289698, −4.75297579295402906626388701752, −3.59889789409010164205295722526, −3.39919770862336868549693868751, −3.27145977687743645557013697797, −3.17950822102262844129885461979, −2.20389589143618221305775567119, −1.67368025402566070603883848501, −0.66756306154638102591564443936,
0.66756306154638102591564443936, 1.67368025402566070603883848501, 2.20389589143618221305775567119, 3.17950822102262844129885461979, 3.27145977687743645557013697797, 3.39919770862336868549693868751, 3.59889789409010164205295722526, 4.75297579295402906626388701752, 5.16220767501999316703122289698, 5.51683083249260160026979033807, 5.54146860361022471232453573477, 5.65956700888107471052541921644, 6.02985952146808533006216944611, 6.81842044350380925908750047663, 7.11378647964160262014460832949, 7.14203736574820061046078816135, 7.34476711637577239965022710427, 8.000783233667318446666613834148, 8.009373213722982671850439819836, 8.228199031050160078963874641245, 8.868877961939504159867187409771, 8.969596023014252945070630651690, 9.002827187948285875855316348389, 9.404360551564499699705224351273, 9.559794286102011012517227700707
