| L(s) = 1 | + 8·5-s + 4·9-s + 20·25-s − 24·29-s + 32·45-s + 28·49-s + 24·53-s + 16·73-s + 7·81-s + 32·97-s − 8·101-s + 40·121-s − 40·125-s + 127-s + 131-s + 137-s + 139-s − 192·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 12·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | + 3.57·5-s + 4/3·9-s + 4·25-s − 4.45·29-s + 4.77·45-s + 4·49-s + 3.29·53-s + 1.87·73-s + 7/9·81-s + 3.24·97-s − 0.796·101-s + 3.63·121-s − 3.57·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 15.9·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.923·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(10.10620409\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.10620409\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 11 | $C_2^2$ | \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 28 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 + 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 54 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 76 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 66 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 59 | $C_2^2$ | \( ( 1 - 20 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 + 124 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 102 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 150 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 4 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 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
−6.79484301747638934233045093316, −6.46733200919290166575572695131, −6.27385470298349342703585772996, −5.89088120566121463826772902796, −5.80302727599945216287442985702, −5.57557092278598495330193474263, −5.54780159970249674670497524613, −5.45234558864586665058978879693, −5.34768188599883540857029705414, −4.83013584178370399785227651657, −4.47301969980647310622541193383, −4.30217931315052869576873636788, −4.01416508615682170033796390267, −3.77912332567335314500394543755, −3.72920245561915219828872716817, −3.27916131719965039637562337335, −3.03030740891582115868814638726, −2.28726654669050915849667372168, −2.24802957359246435592661474010, −2.17116407671517581217757924187, −2.03312369310078930700580700997, −1.80357524413558522450437031713, −1.36495372668161882719190326341, −0.977936771687180912640439980579, −0.53376406031675919981564104269,
0.53376406031675919981564104269, 0.977936771687180912640439980579, 1.36495372668161882719190326341, 1.80357524413558522450437031713, 2.03312369310078930700580700997, 2.17116407671517581217757924187, 2.24802957359246435592661474010, 2.28726654669050915849667372168, 3.03030740891582115868814638726, 3.27916131719965039637562337335, 3.72920245561915219828872716817, 3.77912332567335314500394543755, 4.01416508615682170033796390267, 4.30217931315052869576873636788, 4.47301969980647310622541193383, 4.83013584178370399785227651657, 5.34768188599883540857029705414, 5.45234558864586665058978879693, 5.54780159970249674670497524613, 5.57557092278598495330193474263, 5.80302727599945216287442985702, 5.89088120566121463826772902796, 6.27385470298349342703585772996, 6.46733200919290166575572695131, 6.79484301747638934233045093316
