| L(s) = 1 | + 2-s − 4-s − 7-s − 3·8-s − 14-s − 16-s + 8·17-s + 5·23-s + 3·25-s + 28-s + 4·31-s + 5·32-s + 8·34-s + 10·41-s + 5·46-s − 3·47-s − 11·49-s + 3·50-s + 3·56-s + 4·62-s + 7·64-s − 8·68-s + 9·71-s + 6·73-s − 13·79-s + 10·82-s + 89-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 0.377·7-s − 1.06·8-s − 0.267·14-s − 1/4·16-s + 1.94·17-s + 1.04·23-s + 3/5·25-s + 0.188·28-s + 0.718·31-s + 0.883·32-s + 1.37·34-s + 1.56·41-s + 0.737·46-s − 0.437·47-s − 1.57·49-s + 0.424·50-s + 0.400·56-s + 0.508·62-s + 7/8·64-s − 0.970·68-s + 1.06·71-s + 0.702·73-s − 1.46·79-s + 1.10·82-s + 0.105·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46656 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.565973114\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.565973114\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 23 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 5 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 40 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 48 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 77 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 72 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.931979155588920179444416091271, −9.730858628909913013166985180041, −9.237485680741373320363231393713, −8.586927992739618917631173841862, −8.127249720426935873072997565422, −7.54277458029911124721638165804, −6.91150327572607072805489580760, −6.20101576894892608573310206294, −5.81663468069169815703607315953, −5.06278828992168301966721778008, −4.75361520303320831311554809418, −3.82711369123733498398178151470, −3.27559144041420407146087832075, −2.69946353064010548437610398849, −1.06241843384752710115880621991,
1.06241843384752710115880621991, 2.69946353064010548437610398849, 3.27559144041420407146087832075, 3.82711369123733498398178151470, 4.75361520303320831311554809418, 5.06278828992168301966721778008, 5.81663468069169815703607315953, 6.20101576894892608573310206294, 6.91150327572607072805489580760, 7.54277458029911124721638165804, 8.127249720426935873072997565422, 8.586927992739618917631173841862, 9.237485680741373320363231393713, 9.730858628909913013166985180041, 9.931979155588920179444416091271
