| L(s) = 1 | − 256·4-s + 4.03e3·9-s + 4.91e4·16-s − 1.56e5·25-s − 1.03e6·36-s + 2.95e6·49-s + 8.26e6·61-s − 8.38e6·64-s + 1.15e7·81-s + 4.00e7·100-s − 1.85e7·109-s − 7.79e7·121-s + 127-s + 131-s + 137-s + 139-s + 1.98e8·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2.50e8·169-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | − 2·4-s + 1.84·9-s + 3·16-s − 2·25-s − 3.69·36-s + 3.58·49-s + 4.66·61-s − 4·64-s + 2.40·81-s + 4·100-s − 1.37·109-s − 4·121-s + 5.53·144-s + 4·169-s − 7.17·196-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12960000 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12960000 ^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.749858607\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.749858607\) |
| \(L(\frac{9}{2})\) |
|
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 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{2} \) |
| 3 | $C_2^2$ | \( 1 - 4036 T^{2} + p^{14} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{2} \) |
| good | 7 | $C_2^2$ | \( ( 1 - 1477724 T^{2} + p^{14} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{4} \) |
| 13 | $C_2$ | \( ( 1 - p^{7} T^{2} )^{4} \) |
| 17 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{4} \) |
| 19 | $C_2$ | \( ( 1 - p^{7} T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 + 3270361204 T^{2} + p^{14} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 220254 T + p^{7} T^{2} )^{2}( 1 + 220254 T + p^{7} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - p^{7} T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 - p^{7} T^{2} )^{4} \) |
| 41 | $C_2$ | \( ( 1 - 705588 T + p^{7} T^{2} )^{2}( 1 + 705588 T + p^{7} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 524273282996 T^{2} + p^{14} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 297461652764 T^{2} + p^{14} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{4} \) |
| 59 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{4} \) |
| 61 | $C_2$ | \( ( 1 - 2066408 T + p^{7} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + 10467333640396 T^{2} + p^{14} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 - p^{7} T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 - p^{7} T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 10105170602764 T^{2} + p^{14} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 10220106 T + p^{7} T^{2} )^{2}( 1 + 10220106 T + p^{7} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - p^{7} T^{2} )^{4} \) |
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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
−9.762615568829751535217410543569, −9.257319360515262595497156251092, −9.255696093440928222501617973445, −8.789533115045221305519843394940, −8.460284837091223940465175871765, −8.107838655492538424930106435591, −7.81493250106482919053498049129, −7.55690148324239126632176760599, −7.04207387441168133418402236252, −6.94394630474042811204268409224, −6.32846748091430707971644625287, −5.88291422092854531676122463242, −5.48573836832308187877161909341, −5.21106089385872405784497876946, −4.96975545281064945655345432793, −4.24486096157320172886543790177, −3.99063436843774772533892647281, −3.95165573606613311772468015633, −3.67349514611112678631746563594, −2.76162451783824818017427087424, −2.25440346679169746536244799719, −1.68758322008302963654149323110, −1.15412351157139972087494010303, −0.70445189857397091920354722993, −0.41297726795358583491232952022,
0.41297726795358583491232952022, 0.70445189857397091920354722993, 1.15412351157139972087494010303, 1.68758322008302963654149323110, 2.25440346679169746536244799719, 2.76162451783824818017427087424, 3.67349514611112678631746563594, 3.95165573606613311772468015633, 3.99063436843774772533892647281, 4.24486096157320172886543790177, 4.96975545281064945655345432793, 5.21106089385872405784497876946, 5.48573836832308187877161909341, 5.88291422092854531676122463242, 6.32846748091430707971644625287, 6.94394630474042811204268409224, 7.04207387441168133418402236252, 7.55690148324239126632176760599, 7.81493250106482919053498049129, 8.107838655492538424930106435591, 8.460284837091223940465175871765, 8.789533115045221305519843394940, 9.255696093440928222501617973445, 9.257319360515262595497156251092, 9.762615568829751535217410543569
