| L(s) = 1 | + 4·5-s + 2·13-s + 2·17-s + 6·25-s − 2·29-s + 2·37-s + 2·41-s + 49-s − 2·53-s + 8·65-s − 2·73-s + 8·85-s + 2·89-s + 2·97-s − 4·101-s + 2·109-s − 2·113-s + 2·121-s + 127-s + 131-s + 137-s + 139-s − 8·145-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | + 4·5-s + 2·13-s + 2·17-s + 6·25-s − 2·29-s + 2·37-s + 2·41-s + 49-s − 2·53-s + 8·65-s − 2·73-s + 8·85-s + 2·89-s + 2·97-s − 4·101-s + 2·109-s − 2·113-s + 2·121-s + 127-s + 131-s + 137-s + 139-s − 8·145-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(5.485887775\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.485887775\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| good | 5 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
| 11 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 19 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 23 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 29 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 43 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 53 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 83 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 89 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
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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
−6.21746511288709485677171394781, −5.97362816935965176699709774405, −5.93125340005176059779962328998, −5.91378962935762865518366610226, −5.72985335025634246474894284283, −5.47457360067674072990355435016, −5.23622848318525716016595415803, −5.18187938329127415530572722742, −5.04427850440481505862219094649, −4.38336218551121490118980430582, −4.24274434810113407003037279543, −4.12976162721740679816401452201, −4.08011534403630535356001535356, −3.45945826878210643824807223350, −3.30950899098424419667728438376, −3.17377534971714774847937605544, −3.04024375250871161177008710990, −2.40337941388952180740836899914, −2.30700318763082098220377092855, −2.27283431008472217328534827076, −1.86144454536806154846744435504, −1.71885085858526978628349529123, −1.30117087454715303637527662235, −1.14135811186556960384391995694, −1.02919977329600778195907393244,
1.02919977329600778195907393244, 1.14135811186556960384391995694, 1.30117087454715303637527662235, 1.71885085858526978628349529123, 1.86144454536806154846744435504, 2.27283431008472217328534827076, 2.30700318763082098220377092855, 2.40337941388952180740836899914, 3.04024375250871161177008710990, 3.17377534971714774847937605544, 3.30950899098424419667728438376, 3.45945826878210643824807223350, 4.08011534403630535356001535356, 4.12976162721740679816401452201, 4.24274434810113407003037279543, 4.38336218551121490118980430582, 5.04427850440481505862219094649, 5.18187938329127415530572722742, 5.23622848318525716016595415803, 5.47457360067674072990355435016, 5.72985335025634246474894284283, 5.91378962935762865518366610226, 5.93125340005176059779962328998, 5.97362816935965176699709774405, 6.21746511288709485677171394781
