| L(s) = 1 | − 8·7-s + 6·11-s + 20·13-s + 76·19-s + 24·23-s + 5·25-s + 60·29-s + 4·31-s + 56·37-s + 90·41-s + 22·43-s + 204·47-s + 54·49-s − 174·59-s + 44·61-s − 14·67-s − 148·73-s − 48·77-s + 160·79-s − 312·83-s − 160·91-s + 2·97-s + 444·101-s + 172·103-s − 400·109-s − 24·113-s − 41·121-s + ⋯ |
| L(s) = 1 | − 8/7·7-s + 6/11·11-s + 1.53·13-s + 4·19-s + 1.04·23-s + 1/5·25-s + 2.06·29-s + 4/31·31-s + 1.51·37-s + 2.19·41-s + 0.511·43-s + 4.34·47-s + 1.10·49-s − 2.94·59-s + 0.721·61-s − 0.208·67-s − 2.02·73-s − 0.623·77-s + 2.02·79-s − 3.75·83-s − 1.75·91-s + 2/97·97-s + 4.39·101-s + 1.66·103-s − 3.66·109-s − 0.212·113-s − 0.338·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(14.82144712\) |
| \(L(\frac12)\) |
\(\approx\) |
\(14.82144712\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| good | 7 | $D_4\times C_2$ | \( 1 + 8 T + 10 T^{2} - 352 T^{3} - 2621 T^{4} - 352 p^{2} T^{5} + 10 p^{4} T^{6} + 8 p^{6} T^{7} + p^{8} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 6 T + 7 p T^{2} - 390 T^{3} - 8964 T^{4} - 390 p^{2} T^{5} + 7 p^{5} T^{6} - 6 p^{6} T^{7} + p^{8} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 20 T + 22 T^{2} - 800 T^{3} + 46723 T^{4} - 800 p^{2} T^{5} + 22 p^{4} T^{6} - 20 p^{6} T^{7} + p^{8} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 790 T^{2} + 320907 T^{4} - 790 p^{4} T^{6} + p^{8} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 2 p T + 1023 T^{2} - 2 p^{3} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 24 T + 578 T^{2} - 9264 T^{3} - 29277 T^{4} - 9264 p^{2} T^{5} + 578 p^{4} T^{6} - 24 p^{6} T^{7} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 60 T + 2462 T^{2} - 75720 T^{3} + 1894563 T^{4} - 75720 p^{2} T^{5} + 2462 p^{4} T^{6} - 60 p^{6} T^{7} + p^{8} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 4 T + 1030 T^{2} + 11744 T^{3} + 89299 T^{4} + 11744 p^{2} T^{5} + 1030 p^{4} T^{6} - 4 p^{6} T^{7} + p^{8} T^{8} \) |
| 37 | $C_2$ | \( ( 1 - 14 T + p^{2} T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 - 45 T + 2356 T^{2} - 45 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 22 T - 2375 T^{2} + 18458 T^{3} + 3860164 T^{4} + 18458 p^{2} T^{5} - 2375 p^{4} T^{6} - 22 p^{6} T^{7} + p^{8} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 204 T + 21578 T^{2} - 1572024 T^{3} + 85146003 T^{4} - 1572024 p^{2} T^{5} + 21578 p^{4} T^{6} - 204 p^{6} T^{7} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 9700 T^{2} + 38880102 T^{4} - 9700 p^{4} T^{6} + p^{8} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 174 T + 19397 T^{2} + 1619070 T^{3} + 109595916 T^{4} + 1619070 p^{2} T^{5} + 19397 p^{4} T^{6} + 174 p^{6} T^{7} + p^{8} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 44 T - 5450 T^{2} + 2464 T^{3} + 33503299 T^{4} + 2464 p^{2} T^{5} - 5450 p^{4} T^{6} - 44 p^{6} T^{7} + p^{8} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 14 T - 191 T^{2} - 120274 T^{3} - 20881196 T^{4} - 120274 p^{2} T^{5} - 191 p^{4} T^{6} + 14 p^{6} T^{7} + p^{8} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 9148 T^{2} + 46550598 T^{4} - 9148 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 74 T + 11487 T^{2} + 74 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 - 160 T + 7258 T^{2} - 937600 T^{3} + 137709283 T^{4} - 937600 p^{2} T^{5} + 7258 p^{4} T^{6} - 160 p^{6} T^{7} + p^{8} T^{8} \) |
| 83 | $C_2^2$ | \( ( 1 + 156 T + 15001 T^{2} + 156 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 22468 T^{2} + 231780678 T^{4} - 22468 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 2 T - 8675 T^{2} + 20278 T^{3} - 13241876 T^{4} + 20278 p^{2} T^{5} - 8675 p^{4} T^{6} - 2 p^{6} T^{7} + p^{8} T^{8} \) |
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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
−6.23815531205974755219894868323, −6.19059065796740037700908590705, −5.70671187419513817706726543896, −5.66108020847162495790946867432, −5.64331975652631695699335635138, −5.21458186493688662759400567399, −5.10938578662391252640132466635, −4.77185458418120338753507080011, −4.32392219935286735565107297109, −4.28087313414391594029167260304, −4.26676860046853419802175643567, −3.81902219545204855343213895485, −3.68675188433559511462929770764, −3.22910799283999924091780783616, −3.11702518747926252298455180037, −2.95566761813608632407561024938, −2.83693312245337143217560867850, −2.46351873647662955124014954851, −2.37554572481287018688972237516, −1.70210117489154571244647799554, −1.18156326639398668372176884846, −1.17707553305712283958066325555, −1.05029815419755575348635292947, −0.60777361732385959559167709955, −0.56399570451897981737189089946,
0.56399570451897981737189089946, 0.60777361732385959559167709955, 1.05029815419755575348635292947, 1.17707553305712283958066325555, 1.18156326639398668372176884846, 1.70210117489154571244647799554, 2.37554572481287018688972237516, 2.46351873647662955124014954851, 2.83693312245337143217560867850, 2.95566761813608632407561024938, 3.11702518747926252298455180037, 3.22910799283999924091780783616, 3.68675188433559511462929770764, 3.81902219545204855343213895485, 4.26676860046853419802175643567, 4.28087313414391594029167260304, 4.32392219935286735565107297109, 4.77185458418120338753507080011, 5.10938578662391252640132466635, 5.21458186493688662759400567399, 5.64331975652631695699335635138, 5.66108020847162495790946867432, 5.70671187419513817706726543896, 6.19059065796740037700908590705, 6.23815531205974755219894868323
