| L(s) = 1 | − 8·53-s − 4·81-s + 8·113-s + 8·121-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 + ⋯ |
| L(s) = 1 | − 8·53-s − 4·81-s + 8·113-s + 8·121-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{8} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{8} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.058637272\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.058637272\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T^{8} \) |
| 7 | \( 1 \) |
| good | 3 | \( ( 1 + T^{4} )^{4} \) |
| 11 | \( ( 1 - T )^{8}( 1 + T )^{8} \) |
| 13 | \( ( 1 + T^{8} )^{2} \) |
| 17 | \( ( 1 + T^{8} )^{2} \) |
| 19 | \( ( 1 - T )^{8}( 1 + T )^{8} \) |
| 23 | \( ( 1 + T^{4} )^{4} \) |
| 29 | \( ( 1 + T^{4} )^{4} \) |
| 31 | \( ( 1 + T^{2} )^{8} \) |
| 37 | \( ( 1 + T^{4} )^{4} \) |
| 41 | \( ( 1 + T^{8} )^{2} \) |
| 43 | \( ( 1 + T^{4} )^{4} \) |
| 47 | \( ( 1 + T^{4} )^{4} \) |
| 53 | \( ( 1 + T )^{8}( 1 + T^{2} )^{4} \) |
| 59 | \( ( 1 - T )^{8}( 1 + T )^{8} \) |
| 61 | \( ( 1 + T^{8} )^{2} \) |
| 67 | \( ( 1 + T^{4} )^{4} \) |
| 71 | \( ( 1 - T )^{8}( 1 + T )^{8} \) |
| 73 | \( ( 1 + T^{8} )^{2} \) |
| 79 | \( ( 1 + T^{2} )^{8} \) |
| 83 | \( ( 1 + T^{4} )^{4} \) |
| 89 | \( ( 1 + T^{8} )^{2} \) |
| 97 | \( ( 1 + T^{8} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.65415354768336566974724650499, −3.43088530351153404625137716452, −3.35181076025225690929464859838, −3.28859740290677663932367330546, −3.28656823967365564178585045849, −3.21721838160645157858781019448, −3.07132741715043310461655784990, −2.91223489637797268062145052274, −2.79055536947196672104633573995, −2.76501105465121453502921838981, −2.66725119416108736071871952287, −2.36564819505924779190991791175, −2.24712535752377705102801524397, −2.09412556922186869735400329816, −1.95123329799602626593511518571, −1.83402563255309843185158657600, −1.71760481812844285265734726862, −1.66954332889816148973458914029, −1.58313740405398181243232191503, −1.52966162116759110982921459493, −1.15968296798081494997070088521, −0.872987649445326700278638572938, −0.860143963808678002473206125729, −0.65757041891722221763918326068, −0.25515624994551923654888876164,
0.25515624994551923654888876164, 0.65757041891722221763918326068, 0.860143963808678002473206125729, 0.872987649445326700278638572938, 1.15968296798081494997070088521, 1.52966162116759110982921459493, 1.58313740405398181243232191503, 1.66954332889816148973458914029, 1.71760481812844285265734726862, 1.83402563255309843185158657600, 1.95123329799602626593511518571, 2.09412556922186869735400329816, 2.24712535752377705102801524397, 2.36564819505924779190991791175, 2.66725119416108736071871952287, 2.76501105465121453502921838981, 2.79055536947196672104633573995, 2.91223489637797268062145052274, 3.07132741715043310461655784990, 3.21721838160645157858781019448, 3.28656823967365564178585045849, 3.28859740290677663932367330546, 3.35181076025225690929464859838, 3.43088530351153404625137716452, 3.65415354768336566974724650499
Plot not available for L-functions of degree greater than 10.
