| L(s) = 1 | + 4-s − 2·17-s + 2·29-s + 6·41-s + 2·53-s − 4·61-s − 64-s − 2·68-s + 4·73-s + 81-s − 2·89-s + 2·109-s + 4·113-s + 2·116-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 6·164-s + 167-s − 2·169-s + 173-s + ⋯ |
| L(s) = 1 | + 4-s − 2·17-s + 2·29-s + 6·41-s + 2·53-s − 4·61-s − 64-s − 2·68-s + 4·73-s + 81-s − 2·89-s + 2·109-s + 4·113-s + 2·116-s − 4·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 6·164-s + 167-s − 2·169-s + 173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{8} \cdot 37^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.483325229\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.483325229\) |
| \(L(1)\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | | \( 1 \) |
| 37 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| good | 3 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 7 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 11 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 19 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 29 | $C_2$$\times$$C_2^2$ | \( ( 1 - T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 31 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 - T + T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 53 | $C_2$$\times$$C_2^2$ | \( ( 1 - T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 59 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 61 | $C_1$$\times$$C_2^2$ | \( ( 1 + T )^{4}( 1 - T^{2} + T^{4} ) \) |
| 67 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \) |
| 79 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 83 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
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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
−6.16630630524096794131161258355, −6.14471679369044540232954315180, −5.88631361975778330899983581780, −5.85914437662118805665244560444, −5.42308971380183987536545571423, −5.14000427105691417025556327076, −5.11308147018239799802673424525, −4.63403413753255771279975077233, −4.54423795827166947729060764740, −4.51213308020335398300577713798, −4.28419056230916022995080388534, −3.97613326194533055134601868448, −3.76578334414380738175481936607, −3.73479125326380864409426117049, −3.04665899878209660345667356242, −3.04606007709590653142412796751, −2.81413482607366074624121339362, −2.58480761748897134067001321638, −2.38937685184932740618524699464, −2.12913942530925576557983242255, −2.03084534267029494748203269663, −1.71114282316159398577028843638, −1.06564558925585274278139865091, −1.01569777304532563367244706446, −0.63715488049539210376889932840,
0.63715488049539210376889932840, 1.01569777304532563367244706446, 1.06564558925585274278139865091, 1.71114282316159398577028843638, 2.03084534267029494748203269663, 2.12913942530925576557983242255, 2.38937685184932740618524699464, 2.58480761748897134067001321638, 2.81413482607366074624121339362, 3.04606007709590653142412796751, 3.04665899878209660345667356242, 3.73479125326380864409426117049, 3.76578334414380738175481936607, 3.97613326194533055134601868448, 4.28419056230916022995080388534, 4.51213308020335398300577713798, 4.54423795827166947729060764740, 4.63403413753255771279975077233, 5.11308147018239799802673424525, 5.14000427105691417025556327076, 5.42308971380183987536545571423, 5.85914437662118805665244560444, 5.88631361975778330899983581780, 6.14471679369044540232954315180, 6.16630630524096794131161258355
