| L(s) = 1 | + 16-s + 4·25-s + 4·67-s − 4·79-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 + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯ |
| L(s) = 1 | + 16-s + 4·25-s + 4·67-s − 4·79-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 + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.385762966\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.385762966\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 11 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 19 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 53 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 61 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
| 71 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 73 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 97 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
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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
−6.13133776019401321388510169051, −5.92555473152344411233792404851, −5.78296006982442827819035302821, −5.55704033591448449378196093489, −5.42421416350708734240397328349, −5.12433118170940068414114745194, −4.87394311557662320559463637253, −4.86811669000795365775407743210, −4.68668345871981838052957328624, −4.43821875281113136516768523565, −4.04203104024629401648405164716, −3.93528692086243315427971413839, −3.88407554631482228822492648406, −3.38336366409589532500310773252, −3.11028620019379152938681199562, −3.09662523196564915871590242491, −3.03253884671891232351926797042, −2.60679355840232943341171392509, −2.40539446366987347553481715892, −2.08183176202678648937278717595, −1.87254659880856011903497559680, −1.44785938348991474911451193446, −1.15120527358256937590115026030, −0.951767437125475136559348898996, −0.65498837616697276601980948779,
0.65498837616697276601980948779, 0.951767437125475136559348898996, 1.15120527358256937590115026030, 1.44785938348991474911451193446, 1.87254659880856011903497559680, 2.08183176202678648937278717595, 2.40539446366987347553481715892, 2.60679355840232943341171392509, 3.03253884671891232351926797042, 3.09662523196564915871590242491, 3.11028620019379152938681199562, 3.38336366409589532500310773252, 3.88407554631482228822492648406, 3.93528692086243315427971413839, 4.04203104024629401648405164716, 4.43821875281113136516768523565, 4.68668345871981838052957328624, 4.86811669000795365775407743210, 4.87394311557662320559463637253, 5.12433118170940068414114745194, 5.42421416350708734240397328349, 5.55704033591448449378196093489, 5.78296006982442827819035302821, 5.92555473152344411233792404851, 6.13133776019401321388510169051
