| L(s) = 1 | + 2-s + 3-s + 2·4-s + 6-s − 6·11-s + 2·12-s + 2·13-s + 8·17-s − 4·19-s − 6·22-s − 10·23-s + 2·26-s − 8·29-s − 11·32-s − 6·33-s + 8·34-s + 15·37-s − 4·38-s + 2·39-s − 4·43-s − 12·44-s − 10·46-s + 2·47-s + 12·49-s + 8·51-s + 4·52-s + 5·53-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 4-s + 0.408·6-s − 1.80·11-s + 0.577·12-s + 0.554·13-s + 1.94·17-s − 0.917·19-s − 1.27·22-s − 2.08·23-s + 0.392·26-s − 1.48·29-s − 1.94·32-s − 1.04·33-s + 1.37·34-s + 2.46·37-s − 0.648·38-s + 0.320·39-s − 0.609·43-s − 1.80·44-s − 1.47·46-s + 0.291·47-s + 12/7·49-s + 1.12·51-s + 0.554·52-s + 0.686·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.137780352\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.137780352\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 5 | | \( 1 \) |
| good | 2 | $C_4\times C_2$ | \( 1 - T - T^{2} + 3 T^{3} - T^{4} + 3 p T^{5} - p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $C_2^2$ | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2:C_4$ | \( 1 + 6 T + 5 T^{2} - 6 T^{3} + 49 T^{4} - 6 p T^{5} + 5 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2^2:C_4$ | \( 1 - 2 T + 11 T^{2} - 16 T^{3} + 49 T^{4} - 16 p T^{5} + 11 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^2:C_4$ | \( 1 - 8 T + p T^{2} - 60 T^{3} + 461 T^{4} - 60 p T^{5} + p^{3} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2^2:C_4$ | \( 1 + 4 T - 3 T^{2} + 62 T^{3} + 605 T^{4} + 62 p T^{5} - 3 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_4\times C_2$ | \( 1 + 10 T + 37 T^{2} + 200 T^{3} + 1389 T^{4} + 200 p T^{5} + 37 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2^2:C_4$ | \( 1 + 8 T + 5 T^{2} - 8 p T^{3} - 59 p T^{4} - 8 p^{2} T^{5} + 5 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2^2:C_4$ | \( 1 + 9 T^{2} - 110 T^{3} + 741 T^{4} - 110 p T^{5} + 9 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $C_2^2:C_4$ | \( 1 - 15 T + 63 T^{2} - 65 T^{3} + 144 T^{4} - 65 p T^{5} + 63 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^2:C_4$ | \( 1 - 31 T^{2} + 180 T^{3} + 1501 T^{4} + 180 p T^{5} - 31 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 2 T + 42 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2:C_4$ | \( 1 - 2 T - 23 T^{2} - 250 T^{3} + 2601 T^{4} - 250 p T^{5} - 23 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^2:C_4$ | \( 1 - 5 T - 43 T^{2} + 5 p T^{3} + 1244 T^{4} + 5 p^{2} T^{5} - 43 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_4\times C_2$ | \( 1 - 4 T - 43 T^{2} + 408 T^{3} + 905 T^{4} + 408 p T^{5} - 43 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2:C_4$ | \( 1 - 2 T - 57 T^{2} - 64 T^{3} + 3905 T^{4} - 64 p T^{5} - 57 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2^2:C_4$ | \( 1 + 2 T - 43 T^{2} + 370 T^{3} + 5041 T^{4} + 370 p T^{5} - 43 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2^2:C_4$ | \( 1 - 8 T - 47 T^{2} + 434 T^{3} + 1365 T^{4} + 434 p T^{5} - 47 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2^2:C_4$ | \( 1 - 10 T + 27 T^{2} - 740 T^{3} + 11429 T^{4} - 740 p T^{5} + 27 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_4\times C_2$ | \( 1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8} \) |
| 83 | $C_2^2:C_4$ | \( 1 - 18 T + 41 T^{2} + 1416 T^{3} - 18731 T^{4} + 1416 p T^{5} + 41 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2^2:C_4$ | \( 1 + 9 T - 43 T^{2} - 993 T^{3} - 4460 T^{4} - 993 p T^{5} - 43 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2^2:C_4$ | \( 1 + 2 T - 33 T^{2} + 820 T^{3} + 10061 T^{4} + 820 p T^{5} - 33 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.103092634282163162336186619527, −8.039800154780696598650359868363, −7.83224391203375347823281339148, −7.51827264882841634744818982390, −7.34565977069174786116046211255, −6.99268597979981460579800162028, −6.74450059422270821167302532168, −6.49701876756438252216868774570, −6.00843502614321834326984573606, −5.75762819200125097081027111976, −5.70710269430725567495492238378, −5.66740976391047744465236655478, −5.20159845020355634082914399363, −4.95663508661845574926060382666, −4.46787829613409200990459543090, −4.05969759060182111093038676770, −3.90058934184434189992978776609, −3.60608836262713645447694228499, −3.38486713852217235069959721210, −2.84525683551109382577733884676, −2.61239408439806242533916849068, −2.29712323887174060236925048768, −2.03586946552576355696685847344, −1.49452257119849312698406121408, −0.56630178829666160651123029828,
0.56630178829666160651123029828, 1.49452257119849312698406121408, 2.03586946552576355696685847344, 2.29712323887174060236925048768, 2.61239408439806242533916849068, 2.84525683551109382577733884676, 3.38486713852217235069959721210, 3.60608836262713645447694228499, 3.90058934184434189992978776609, 4.05969759060182111093038676770, 4.46787829613409200990459543090, 4.95663508661845574926060382666, 5.20159845020355634082914399363, 5.66740976391047744465236655478, 5.70710269430725567495492238378, 5.75762819200125097081027111976, 6.00843502614321834326984573606, 6.49701876756438252216868774570, 6.74450059422270821167302532168, 6.99268597979981460579800162028, 7.34565977069174786116046211255, 7.51827264882841634744818982390, 7.83224391203375347823281339148, 8.039800154780696598650359868363, 8.103092634282163162336186619527
