| L(s) = 1 | − 2·4-s + 3·16-s + 25-s + 4·31-s − 2·49-s − 4·64-s + 8·79-s − 2·100-s + 2·121-s − 8·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 4·196-s + 197-s + ⋯ |
| L(s) = 1 | − 2·4-s + 3·16-s + 25-s + 4·31-s − 2·49-s − 4·64-s + 8·79-s − 2·100-s + 2·121-s − 8·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 4·196-s + 197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6336752003\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6336752003\) |
| \(L(1)\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| good | 7 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 17 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 29 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 43 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 47 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 53 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 67 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 79 | $C_1$ | \( ( 1 - T )^{8} \) |
| 83 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
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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
−7.51778571151360962519223862736, −6.97021515484389462163631641657, −6.74051948876620824484321635825, −6.50430963435345288618608926388, −6.43400850706305980587936080306, −6.19186171240020255060742843792, −6.05106501151413431306618060982, −5.52502600381031228495468223273, −5.49588552536215980422463874785, −5.06209797470060113091615043386, −4.86994752018897555580602482898, −4.79355264425500666774946558942, −4.58426603438368821504978845332, −4.51904233837962069829048957274, −4.01143962587889608455894798016, −3.72825071393392038502542731028, −3.52781953521080997473902996350, −3.32062298491310001628278791586, −3.09842598395990234858796801035, −2.67435219224706951925869975333, −2.24268083273325330541403018118, −2.20834013717454747780806594918, −1.39470754395918547522919556869, −0.984298241188176431903440917282, −0.869459263362982107627160513250,
0.869459263362982107627160513250, 0.984298241188176431903440917282, 1.39470754395918547522919556869, 2.20834013717454747780806594918, 2.24268083273325330541403018118, 2.67435219224706951925869975333, 3.09842598395990234858796801035, 3.32062298491310001628278791586, 3.52781953521080997473902996350, 3.72825071393392038502542731028, 4.01143962587889608455894798016, 4.51904233837962069829048957274, 4.58426603438368821504978845332, 4.79355264425500666774946558942, 4.86994752018897555580602482898, 5.06209797470060113091615043386, 5.49588552536215980422463874785, 5.52502600381031228495468223273, 6.05106501151413431306618060982, 6.19186171240020255060742843792, 6.43400850706305980587936080306, 6.50430963435345288618608926388, 6.74051948876620824484321635825, 6.97021515484389462163631641657, 7.51778571151360962519223862736
