| L(s) = 1 | + 36·3-s − 3.40e3·9-s − 4.62e4·19-s − 2.93e5·25-s − 2.12e5·27-s − 1.98e6·43-s − 1.50e6·49-s − 1.66e6·57-s − 5.60e6·67-s − 8.89e6·73-s − 1.05e7·75-s + 6.37e6·81-s + 2.74e7·97-s + 7.15e7·121-s + 127-s − 7.12e7·129-s + 131-s + 137-s + 139-s − 5.41e7·147-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4.52e7·169-s + 1.57e8·171-s + ⋯ |
| L(s) = 1 | + 0.769·3-s − 1.55·9-s − 1.54·19-s − 3.75·25-s − 2.08·27-s − 3.79·43-s − 1.82·49-s − 1.19·57-s − 2.27·67-s − 2.67·73-s − 2.89·75-s + 1.33·81-s + 3.05·97-s + 3.67·121-s − 2.92·129-s − 1.40·147-s + 0.721·169-s + 2.40·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84934656 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84934656 ^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.5385604933\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5385604933\) |
| \(L(\frac{9}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( ( 1 - 2 p^{2} T + p^{7} T^{2} )^{2} \) |
| good | 5 | $C_2^2$ | \( ( 1 + 5866 p^{2} T^{2} + p^{14} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + 751570 T^{2} + p^{14} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 35789342 T^{2} + p^{14} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 1741490 p T^{2} + p^{14} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 61393730 T^{2} + p^{14} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 11570 T + p^{7} T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 + 3725533390 T^{2} + p^{14} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 31499935018 T^{2} + p^{14} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 49885402622 T^{2} + p^{14} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 80259050 p^{2} T^{2} + p^{14} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 226584602162 T^{2} + p^{14} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 495062 T + p^{7} T^{2} )^{4} \) |
| 47 | $C_2^2$ | \( ( 1 - 277104744290 T^{2} + p^{14} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 2021761791610 T^{2} + p^{14} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 2902252328702 T^{2} + p^{14} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 3094549289642 T^{2} + p^{14} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 1400126 T + p^{7} T^{2} )^{4} \) |
| 71 | $C_2^2$ | \( ( 1 + 5449089705838 T^{2} + p^{14} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 2223598 T + p^{7} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 7153320805022 T^{2} + p^{14} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 45191963757710 T^{2} + p^{14} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 54294758858162 T^{2} + p^{14} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 6867926 T + p^{7} T^{2} )^{4} \) |
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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
−8.803662035707899252174752756311, −8.636386585882752086122057712754, −8.337172658564417872274879303416, −8.061600118158867151724760129549, −7.68894754730137574415802784267, −7.50236661449081021098977209712, −7.22930475705132557905327978913, −6.61633567047793226136889426074, −6.15504558459812454129726293275, −6.10788173009779909190337495953, −6.00482654465997007927742548066, −5.45760429858590486212225439408, −5.08538470560778546916329153949, −4.62412682558281947705684209117, −4.43396230107146571270478759726, −3.80308062409237325703240677046, −3.56505535258397094000711449875, −3.22718521945595389105221681005, −2.98504760399756537303212813644, −2.30376178945926633369953622321, −2.04179064360144430223543871959, −1.77218522410039202000093701494, −1.41791723942763422291340263866, −0.25862422609195787910063983336, −0.24506555241710578022399184665,
0.24506555241710578022399184665, 0.25862422609195787910063983336, 1.41791723942763422291340263866, 1.77218522410039202000093701494, 2.04179064360144430223543871959, 2.30376178945926633369953622321, 2.98504760399756537303212813644, 3.22718521945595389105221681005, 3.56505535258397094000711449875, 3.80308062409237325703240677046, 4.43396230107146571270478759726, 4.62412682558281947705684209117, 5.08538470560778546916329153949, 5.45760429858590486212225439408, 6.00482654465997007927742548066, 6.10788173009779909190337495953, 6.15504558459812454129726293275, 6.61633567047793226136889426074, 7.22930475705132557905327978913, 7.50236661449081021098977209712, 7.68894754730137574415802784267, 8.061600118158867151724760129549, 8.337172658564417872274879303416, 8.636386585882752086122057712754, 8.803662035707899252174752756311
