| L(s) = 1 | + 483·4-s + 4.30e4·11-s + 1.71e5·16-s + 1.91e5·19-s − 5.35e6·29-s + 2.15e7·31-s − 5.21e7·41-s + 2.07e7·44-s + 7.42e6·49-s − 7.09e7·59-s + 6.82e8·61-s + 1.79e8·64-s − 4.20e8·71-s + 9.26e7·76-s + 4.95e7·79-s − 8.55e8·89-s − 2.71e7·101-s − 8.79e9·109-s − 2.58e9·116-s − 2.19e9·121-s + 1.04e10·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 0.943·4-s + 0.886·11-s + 0.655·16-s + 0.337·19-s − 1.40·29-s + 4.19·31-s − 2.88·41-s + 0.835·44-s + 0.183·49-s − 0.762·59-s + 6.31·61-s + 1.34·64-s − 1.96·71-s + 0.318·76-s + 0.143·79-s − 1.44·89-s − 0.0260·101-s − 5.97·109-s − 1.32·116-s − 0.930·121-s + 3.95·124-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(10-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(5.537584749\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.537584749\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $D_4\times C_2$ | \( 1 - 483 T^{2} + 15377 p^{2} T^{4} - 483 p^{18} T^{6} + p^{36} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 151516 p^{2} T^{2} - 894549623322 p^{4} T^{4} - 151516 p^{20} T^{6} + p^{36} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 21512 T + 1790961254 T^{2} - 21512 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 11151625772 T^{2} + \)\(24\!\cdots\!78\)\( T^{4} - 11151625772 p^{18} T^{6} + p^{36} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 400705004220 T^{2} + \)\(67\!\cdots\!18\)\( T^{4} - 400705004220 p^{18} T^{6} + p^{36} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 95896 T + 629883192438 T^{2} - 95896 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 6185762667356 T^{2} + \)\(15\!\cdots\!18\)\( T^{4} - 6185762667356 p^{18} T^{6} + p^{36} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 2678212 T + 15397908029438 T^{2} + 2678212 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 - 10782432 T + 69294691361342 T^{2} - 10782432 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 235813135792844 T^{2} + \)\(37\!\cdots\!18\)\( T^{4} - 235813135792844 p^{18} T^{6} + p^{36} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 26060372 T + 693239183881142 T^{2} + 26060372 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 1449556389870860 T^{2} + \)\(10\!\cdots\!98\)\( T^{4} - 1449556389870860 p^{18} T^{6} + p^{36} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 233226346198980 T^{2} + \)\(41\!\cdots\!78\)\( T^{4} + 233226346198980 p^{18} T^{6} + p^{36} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 13159896059574156 T^{2} + \)\(65\!\cdots\!18\)\( T^{4} - 13159896059574156 p^{18} T^{6} + p^{36} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 35494664 T + 7388006896329158 T^{2} + 35494664 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 341497340 T + 52053805546777278 T^{2} - 341497340 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 66154849722487660 T^{2} + \)\(25\!\cdots\!18\)\( T^{4} - 66154849722487660 p^{18} T^{6} + p^{36} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 210286064 T + 91549406631588686 T^{2} + 210286064 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 206940304125633116 T^{2} + \)\(17\!\cdots\!78\)\( T^{4} - 206940304125633116 p^{18} T^{6} + p^{36} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 24755040 T + 43090694479668638 T^{2} - 24755040 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 621372529743013292 T^{2} + \)\(16\!\cdots\!98\)\( T^{4} - 621372529743013292 p^{18} T^{6} + p^{36} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 427639116 T + 702028302670151638 T^{2} + 427639116 p^{9} T^{3} + p^{18} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 1275322199616361340 T^{2} + \)\(12\!\cdots\!78\)\( T^{4} - 1275322199616361340 p^{18} T^{6} + p^{36} T^{8} \) |
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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
−7.28717827036026305849675168126, −6.70172413831418240867594130870, −6.65463642935769904517763918732, −6.62463221951468595451294680870, −6.60954092217936931472954875990, −5.85348054807708652702172682892, −5.60221130848197740141412193781, −5.39129848761866319170292941299, −5.32218853520486496006284094366, −4.85988138997877959040394577960, −4.35403605980701204066429355464, −4.22498518057796194892453276407, −4.03975820461242183261308659389, −3.65049290217030525433835964817, −3.13850741834502870872645497744, −3.13218546499582483420605721729, −2.79790887513733812991352392185, −2.35839363456633738760095460839, −2.11231660779699110062834935118, −1.88923808207252739014759445807, −1.40260137467796832910375172178, −1.17274841845702392970279672006, −0.917088700453334449262465056765, −0.62236455422541160129442007362, −0.20271944333776777745827372235,
0.20271944333776777745827372235, 0.62236455422541160129442007362, 0.917088700453334449262465056765, 1.17274841845702392970279672006, 1.40260137467796832910375172178, 1.88923808207252739014759445807, 2.11231660779699110062834935118, 2.35839363456633738760095460839, 2.79790887513733812991352392185, 3.13218546499582483420605721729, 3.13850741834502870872645497744, 3.65049290217030525433835964817, 4.03975820461242183261308659389, 4.22498518057796194892453276407, 4.35403605980701204066429355464, 4.85988138997877959040394577960, 5.32218853520486496006284094366, 5.39129848761866319170292941299, 5.60221130848197740141412193781, 5.85348054807708652702172682892, 6.60954092217936931472954875990, 6.62463221951468595451294680870, 6.65463642935769904517763918732, 6.70172413831418240867594130870, 7.28717827036026305849675168126
