| L(s) = 1 | + 8·23-s − 2·25-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-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 + 241-s + 251-s + 257-s + ⋯ |
| L(s) = 1 | + 8·23-s − 2·25-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-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 + 241-s + 251-s + 257-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{8} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.154265401\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.154265401\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 7 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 23 | $C_1$ | \( ( 1 - T )^{8} \) |
| 29 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 47 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 59 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 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_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 79 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 89 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
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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
−6.53856341888398121526241206630, −6.25528272141656758958509754123, −5.97032270251997281946222035171, −5.69946142624731506957614543564, −5.49005516939160698353316654336, −5.34428609002587137181062018068, −5.25347940829994890954191872903, −5.07628913161168378683820048224, −4.73715458753281013145042218267, −4.57717646231359259236740012048, −4.46720627732135103556286167878, −4.14435274004206410912863426062, −4.02928753092474992346755664735, −3.35062120206048695465492417240, −3.30611141443052772614680051194, −3.25913280662840655263764674716, −3.19775863235923938162513071896, −2.72803698089358560208662802568, −2.47256928897597453573065392110, −2.41060109078448395570202238038, −1.75131245771021746880707341781, −1.72186417172226829918787137117, −1.15923148354225909248848091054, −0.973951173388020184084570989698, −0.77878056760116765090875203817,
0.77878056760116765090875203817, 0.973951173388020184084570989698, 1.15923148354225909248848091054, 1.72186417172226829918787137117, 1.75131245771021746880707341781, 2.41060109078448395570202238038, 2.47256928897597453573065392110, 2.72803698089358560208662802568, 3.19775863235923938162513071896, 3.25913280662840655263764674716, 3.30611141443052772614680051194, 3.35062120206048695465492417240, 4.02928753092474992346755664735, 4.14435274004206410912863426062, 4.46720627732135103556286167878, 4.57717646231359259236740012048, 4.73715458753281013145042218267, 5.07628913161168378683820048224, 5.25347940829994890954191872903, 5.34428609002587137181062018068, 5.49005516939160698353316654336, 5.69946142624731506957614543564, 5.97032270251997281946222035171, 6.25528272141656758958509754123, 6.53856341888398121526241206630
