| L(s) = 1 | + 176·4-s − 486·9-s − 5.09e4·13-s + 1.45e4·16-s + 2.27e5·25-s − 8.55e4·36-s + 1.73e5·37-s + 1.29e6·49-s − 8.96e6·52-s + 1.25e6·61-s − 3.15e5·64-s − 1.03e6·73-s − 4.54e6·81-s + 2.89e7·97-s + 4.01e7·100-s + 3.92e7·109-s + 2.47e7·117-s − 7.01e7·121-s + 127-s + 131-s + 137-s + 139-s − 7.09e6·144-s + 3.04e7·148-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | + 11/8·4-s − 2/9·9-s − 6.42·13-s + 0.890·16-s + 2.91·25-s − 0.305·36-s + 0.562·37-s + 1.57·49-s − 8.83·52-s + 0.708·61-s − 0.150·64-s − 0.312·73-s − 0.950·81-s + 3.22·97-s + 4.01·100-s + 2.90·109-s + 1.42·117-s − 3.59·121-s − 0.197·144-s + 0.773·148-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20736 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20736 ^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.567858543\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.567858543\) |
| \(L(\frac{9}{2})\) |
|
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 | $C_2^2$ | \( 1 - 11 p^{4} T^{2} + p^{14} T^{4} \) |
| 3 | $C_2^2$ | \( 1 + 2 p^{5} T^{2} + p^{14} T^{4} \) |
| good | 5 | $C_2^2$ | \( ( 1 - 22786 p T^{2} + p^{14} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 648626 T^{2} + p^{14} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 + 35075542 T^{2} + p^{14} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 12730 T + p^{7} T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 - 1609634 p^{2} T^{2} + p^{14} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 1215969338 T^{2} + p^{14} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 1571942926 T^{2} + p^{14} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 23241291098 T^{2} + p^{14} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 46858094882 T^{2} + p^{14} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 43310 T + p^{7} T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 + 227957477518 T^{2} + p^{14} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 538623749354 T^{2} + p^{14} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 695418926494 T^{2} + p^{14} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 2227661146154 T^{2} + p^{14} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 4402559099638 T^{2} + p^{14} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 314198 T + p^{7} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 11463132309146 T^{2} + p^{14} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 18120013316782 T^{2} + p^{14} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 259270 T + p^{7} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 11004425314178 T^{2} + p^{14} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 47771708513594 T^{2} + p^{14} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 73372933660178 T^{2} + p^{14} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 7243010 T + p^{7} T^{2} )^{4} \) |
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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
−14.32332380056691747118073611979, −13.12695706556131790488637152114, −12.77999850420256112508093841107, −12.50541146134483754784196071845, −12.08737095135689565051681197017, −11.96906524196204569769380771503, −11.61152712056322090738830172542, −10.95904138341729437493402470583, −10.40744884795875416383243501229, −9.996807994387519671235422656562, −9.996554052835943321498598308815, −9.232252792017005568639828688244, −8.918592367736754325859666628811, −7.938236002908821722675550354989, −7.34238542360966431205510754993, −7.23475880578486113861851601606, −7.09175021848274678470365949358, −6.31400819264188877647603214996, −5.26499654883386364413997526171, −4.92190864043536808766193358771, −4.62639712078133131638049165746, −2.86836281567222283482011616891, −2.56429165814071843872503216317, −2.22195042962927585475181372723, −0.46445618493903922016043319982,
0.46445618493903922016043319982, 2.22195042962927585475181372723, 2.56429165814071843872503216317, 2.86836281567222283482011616891, 4.62639712078133131638049165746, 4.92190864043536808766193358771, 5.26499654883386364413997526171, 6.31400819264188877647603214996, 7.09175021848274678470365949358, 7.23475880578486113861851601606, 7.34238542360966431205510754993, 7.938236002908821722675550354989, 8.918592367736754325859666628811, 9.232252792017005568639828688244, 9.996554052835943321498598308815, 9.996807994387519671235422656562, 10.40744884795875416383243501229, 10.95904138341729437493402470583, 11.61152712056322090738830172542, 11.96906524196204569769380771503, 12.08737095135689565051681197017, 12.50541146134483754784196071845, 12.77999850420256112508093841107, 13.12695706556131790488637152114, 14.32332380056691747118073611979
