| L(s) = 1 | − 4·2-s − 6·3-s + 4·4-s + 24·6-s + 4·7-s + 16·8-s + 21·9-s − 12·11-s − 24·12-s − 18·13-s − 16·14-s − 64·16-s + 26·17-s − 84·18-s + 16·19-s − 24·21-s + 48·22-s + 24·23-s − 96·24-s + 94·25-s + 72·26-s − 54·27-s + 16·28-s + 14·29-s − 128·31-s + 64·32-s + 72·33-s + ⋯ |
| L(s) = 1 | − 2·2-s − 2·3-s + 4-s + 4·6-s + 4/7·7-s + 2·8-s + 7/3·9-s − 1.09·11-s − 2·12-s − 1.38·13-s − 8/7·14-s − 4·16-s + 1.52·17-s − 4.66·18-s + 0.842·19-s − 8/7·21-s + 2.18·22-s + 1.04·23-s − 4·24-s + 3.75·25-s + 2.76·26-s − 2·27-s + 4/7·28-s + 0.482·29-s − 4.12·31-s + 2·32-s + 2.18·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.004104378512\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.004104378512\) |
| \(L(2)\) |
|
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 | $C_2$ | \( ( 1 + p T + p^{2} T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 + p T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 18 T + 277 T^{2} + 18 p^{2} T^{3} + p^{4} T^{4} \) |
| good | 5 | $C_2^2$ | \( ( 1 - 47 T^{2} + p^{4} T^{4} )^{2} \) |
| 7 | $D_4\times C_2$ | \( 1 - 4 T + 8 p T^{2} + 552 T^{3} - 1585 T^{4} + 552 p^{2} T^{5} + 8 p^{5} T^{6} - 4 p^{6} T^{7} + p^{8} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 12 T + 8 T^{2} - 1272 T^{3} - 13569 T^{4} - 1272 p^{2} T^{5} + 8 p^{4} T^{6} + 12 p^{6} T^{7} + p^{8} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 26 T + 71 T^{2} - 702 T^{3} + 87140 T^{4} - 702 p^{2} T^{5} + 71 p^{4} T^{6} - 26 p^{6} T^{7} + p^{8} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 16 T - 388 T^{2} + 1248 T^{3} + 191999 T^{4} + 1248 p^{2} T^{5} - 388 p^{4} T^{6} - 16 p^{6} T^{7} + p^{8} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 24 T + 872 T^{2} - 16320 T^{3} + 284127 T^{4} - 16320 p^{2} T^{5} + 872 p^{4} T^{6} - 24 p^{6} T^{7} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 14 T - 967 T^{2} + 7266 T^{3} + 480452 T^{4} + 7266 p^{2} T^{5} - 967 p^{4} T^{6} - 14 p^{6} T^{7} + p^{8} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 64 T + 2378 T^{2} + 64 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 + 42 T + 3047 T^{2} + 103278 T^{3} + 4977492 T^{4} + 103278 p^{2} T^{5} + 3047 p^{4} T^{6} + 42 p^{6} T^{7} + p^{8} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 + 18 T - 337 T^{2} - 8010 T^{3} - 2446188 T^{4} - 8010 p^{2} T^{5} - 337 p^{4} T^{6} + 18 p^{6} T^{7} + p^{8} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 24 T + 3512 T^{2} + 79680 T^{3} + 7958607 T^{4} + 79680 p^{2} T^{5} + 3512 p^{4} T^{6} + 24 p^{6} T^{7} + p^{8} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 - 24 T + 4420 T^{2} - 24 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 53 | $D_{4}$ | \( ( 1 + 50 T + 5675 T^{2} + 50 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 56 T - 345 T^{2} - 56 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 + 54 T - 1705 T^{2} - 152334 T^{3} - 3342636 T^{4} - 152334 p^{2} T^{5} - 1705 p^{4} T^{6} + 54 p^{6} T^{7} + p^{8} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 40 T - 820 T^{2} + 262320 T^{3} - 21477121 T^{4} + 262320 p^{2} T^{5} - 820 p^{4} T^{6} - 40 p^{6} T^{7} + p^{8} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 + 176 T + 14428 T^{2} + 1138016 T^{3} + 92274607 T^{4} + 1138016 p^{2} T^{5} + 14428 p^{4} T^{6} + 176 p^{6} T^{7} + p^{8} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 3502 T^{2} + 10766835 T^{4} - 3502 p^{4} T^{6} + p^{8} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 21460 T^{2} + 192705894 T^{4} - 21460 p^{4} T^{6} + p^{8} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 52 T + 2952 T^{2} + 52 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 288 T + 48698 T^{2} + 6062400 T^{3} + 599360067 T^{4} + 6062400 p^{2} T^{5} + 48698 p^{4} T^{6} + 288 p^{6} T^{7} + p^{8} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 12 T + 12062 T^{2} + 144168 T^{3} + 56258547 T^{4} + 144168 p^{2} T^{5} + 12062 p^{4} T^{6} + 12 p^{6} T^{7} + p^{8} T^{8} \) |
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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
−9.346119711871586466577402721379, −9.156571274984918399808873850794, −8.605205106900896596743154155355, −8.533853373002466376556949623969, −8.272572695995182144950496552876, −7.81220881275431828904238859368, −7.63502620860301644801058246596, −7.08371132230718315262100943490, −7.05893044502211142203515476736, −6.99169966467142168461178006073, −6.93388672732338032229310597976, −5.84458960174305852223744116333, −5.74876655650937212865980544862, −5.31300262739878501519126232347, −5.03016098506055828442844640764, −4.98622931023334668337418878214, −4.87662158218567427390005263995, −4.24093653711262960774603254607, −3.80191343608915189506214415888, −3.01963115412892966195814431213, −2.89389129978056465262671681256, −1.83829709299737051347173211989, −1.32493046098073202421818770677, −1.09284552701455108655210818592, −0.04675203268595427823241594932,
0.04675203268595427823241594932, 1.09284552701455108655210818592, 1.32493046098073202421818770677, 1.83829709299737051347173211989, 2.89389129978056465262671681256, 3.01963115412892966195814431213, 3.80191343608915189506214415888, 4.24093653711262960774603254607, 4.87662158218567427390005263995, 4.98622931023334668337418878214, 5.03016098506055828442844640764, 5.31300262739878501519126232347, 5.74876655650937212865980544862, 5.84458960174305852223744116333, 6.93388672732338032229310597976, 6.99169966467142168461178006073, 7.05893044502211142203515476736, 7.08371132230718315262100943490, 7.63502620860301644801058246596, 7.81220881275431828904238859368, 8.272572695995182144950496552876, 8.533853373002466376556949623969, 8.605205106900896596743154155355, 9.156571274984918399808873850794, 9.346119711871586466577402721379
