L(s) = 1 | + 2-s − 3·3-s − 3·6-s + 2·7-s + 6·9-s + 2·14-s + 6·18-s − 6·21-s − 3·23-s − 10·27-s + 3·29-s − 32-s − 2·41-s − 6·42-s + 2·43-s − 3·46-s − 3·47-s + 49-s − 10·54-s + 3·58-s + 3·61-s + 12·63-s − 64-s + 2·67-s + 9·69-s + 15·81-s − 2·82-s + ⋯ |
L(s) = 1 | + 2-s − 3·3-s − 3·6-s + 2·7-s + 6·9-s + 2·14-s + 6·18-s − 6·21-s − 3·23-s − 10·27-s + 3·29-s − 32-s − 2·41-s − 6·42-s + 2·43-s − 3·46-s − 3·47-s + 49-s − 10·54-s + 3·58-s + 3·61-s + 12·63-s − 64-s + 2·67-s + 9·69-s + 15·81-s − 2·82-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.011234410\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.011234410\) |
\(L(1)\) |
|
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 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 5 | | \( 1 \) |
good | 3 | $C_1$$\times$$C_4$ | \( ( 1 + T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} ) \) |
| 7 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 11 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 13 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 17 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 19 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 23 | $C_1$$\times$$C_4$ | \( ( 1 + T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} ) \) |
| 29 | $C_1$$\times$$C_4$ | \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 31 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 37 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 41 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 43 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 47 | $C_1$$\times$$C_4$ | \( ( 1 + T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} ) \) |
| 53 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 59 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 61 | $C_1$$\times$$C_4$ | \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 67 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 71 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 73 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 79 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 83 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 89 | $C_1$$\times$$C_4$ | \( ( 1 - T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 97 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
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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
−6.53205321172626881922714777426, −6.17404689567621861619154803652, −6.07971831909751252502381221942, −5.98130236904461519546577517603, −5.50004580793099174433454266777, −5.41147387494631502569469772070, −5.24759176283247482483942251059, −5.00487188715650907674742067289, −4.92010890859246411725800940399, −4.78830838677638834278201643373, −4.50343880762503994557043533552, −4.42710052881032456329789226494, −4.30111191556061734465061944626, −3.81165193560217516876460988218, −3.62073683677735157437505673779, −3.54479439617205875246132175120, −3.52081820727396470738447883201, −2.54323362755843291156891669940, −2.41618162202106074767020392860, −2.19302643562404637607102922316, −1.81780882985048552637600785011, −1.50764904966784089992766070636, −1.50566936130216917692412130321, −0.880397737145067512000591798564, −0.59419872341747865676429970734,
0.59419872341747865676429970734, 0.880397737145067512000591798564, 1.50566936130216917692412130321, 1.50764904966784089992766070636, 1.81780882985048552637600785011, 2.19302643562404637607102922316, 2.41618162202106074767020392860, 2.54323362755843291156891669940, 3.52081820727396470738447883201, 3.54479439617205875246132175120, 3.62073683677735157437505673779, 3.81165193560217516876460988218, 4.30111191556061734465061944626, 4.42710052881032456329789226494, 4.50343880762503994557043533552, 4.78830838677638834278201643373, 4.92010890859246411725800940399, 5.00487188715650907674742067289, 5.24759176283247482483942251059, 5.41147387494631502569469772070, 5.50004580793099174433454266777, 5.98130236904461519546577517603, 6.07971831909751252502381221942, 6.17404689567621861619154803652, 6.53205321172626881922714777426