| L(s) = 1 | + 2·3-s + 2·5-s + 9-s + 4·15-s + 16-s + 3·25-s − 2·27-s − 2·31-s − 4·37-s + 2·45-s + 2·47-s + 2·48-s + 4·53-s − 2·59-s + 2·67-s − 4·71-s + 6·75-s + 2·80-s − 4·81-s − 4·93-s − 2·97-s + 2·103-s − 8·111-s + 2·113-s + 6·125-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | + 2·3-s + 2·5-s + 9-s + 4·15-s + 16-s + 3·25-s − 2·27-s − 2·31-s − 4·37-s + 2·45-s + 2·47-s + 2·48-s + 4·53-s − 2·59-s + 2·67-s − 4·71-s + 6·75-s + 2·80-s − 4·81-s − 4·93-s − 2·97-s + 2·103-s − 8·111-s + 2·113-s + 6·125-s + 127-s + 131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.669471208\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.669471208\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 7 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 29 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 47 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
| 59 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 61 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 83 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 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
−7.31867360494467816568380471352, −7.16771593774900244636901159438, −7.10297103897251148556946399910, −6.76905103510706204498182617591, −6.10160206583250533886459593606, −6.05841395278505951749810063999, −5.95632418316677480908896963779, −5.79353587906435914021904395580, −5.60857862348732504522626856422, −5.14571203954136845081634654854, −5.04676289425161766849602400909, −5.01557043382854351366197750403, −4.50045355799031918187645100809, −4.10667743345344212108250949011, −3.70689898853406302105373239916, −3.68889216417039703106138956221, −3.42175591788946648275141168233, −3.29622586802518816694758198894, −2.66182498676302749541908147473, −2.65769916299205601557110762508, −2.45344756700752563808013222718, −1.98457268831581007194767202521, −1.83265225700011097594630289268, −1.53911232748449003147121250377, −1.11762132914441298332085719015,
1.11762132914441298332085719015, 1.53911232748449003147121250377, 1.83265225700011097594630289268, 1.98457268831581007194767202521, 2.45344756700752563808013222718, 2.65769916299205601557110762508, 2.66182498676302749541908147473, 3.29622586802518816694758198894, 3.42175591788946648275141168233, 3.68889216417039703106138956221, 3.70689898853406302105373239916, 4.10667743345344212108250949011, 4.50045355799031918187645100809, 5.01557043382854351366197750403, 5.04676289425161766849602400909, 5.14571203954136845081634654854, 5.60857862348732504522626856422, 5.79353587906435914021904395580, 5.95632418316677480908896963779, 6.05841395278505951749810063999, 6.10160206583250533886459593606, 6.76905103510706204498182617591, 7.10297103897251148556946399910, 7.16771593774900244636901159438, 7.31867360494467816568380471352
