| L(s) = 1 | − 3·2-s − 3-s + 7·4-s + 3·6-s − 3·7-s − 15·8-s − 11-s − 7·12-s + 6·13-s + 9·14-s + 30·16-s − 13·17-s + 10·19-s + 3·21-s + 3·22-s + 26·23-s + 15·24-s − 18·26-s − 21·28-s + 13·31-s − 57·32-s + 33-s + 39·34-s + 2·37-s − 30·38-s − 6·39-s − 7·41-s + ⋯ |
| L(s) = 1 | − 2.12·2-s − 0.577·3-s + 7/2·4-s + 1.22·6-s − 1.13·7-s − 5.30·8-s − 0.301·11-s − 2.02·12-s + 1.66·13-s + 2.40·14-s + 15/2·16-s − 3.15·17-s + 2.29·19-s + 0.654·21-s + 0.639·22-s + 5.42·23-s + 3.06·24-s − 3.53·26-s − 3.96·28-s + 2.33·31-s − 10.0·32-s + 0.174·33-s + 6.68·34-s + 0.328·37-s − 4.86·38-s − 0.960·39-s − 1.09·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.107588679\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.107588679\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 11 | $C_4$ | \( 1 + T + 21 T^{2} + p T^{3} + p^{2} T^{4} \) |
| good | 2 | $C_2^2:C_4$ | \( 1 + 3 T + p T^{2} + T^{4} + p^{3} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $C_4\times C_2$ | \( 1 + 3 T + 2 T^{2} - 15 T^{3} - 59 T^{4} - 15 p T^{5} + 2 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2^2:C_4$ | \( 1 - 6 T + 23 T^{2} - 120 T^{3} + 601 T^{4} - 120 p T^{5} + 23 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^2:C_4$ | \( 1 + 13 T + 52 T^{2} - 25 T^{3} - 729 T^{4} - 25 p T^{5} + 52 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2^2:C_4$ | \( 1 - 10 T + 21 T^{2} + 10 p T^{3} - 1519 T^{4} + 10 p^{2} T^{5} + 21 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 - 13 T + 87 T^{2} - 13 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2:C_4$ | \( 1 - 19 T^{2} - 120 T^{3} + 721 T^{4} - 120 p T^{5} - 19 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $C_4\times C_2$ | \( 1 - 13 T + 33 T^{2} + 379 T^{3} - 3700 T^{4} + 379 p T^{5} + 33 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2^2:C_4$ | \( 1 - 2 T + 27 T^{2} - 160 T^{3} + 1841 T^{4} - 160 p T^{5} + 27 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^2:C_4$ | \( 1 + 7 T - 22 T^{2} - 161 T^{3} + 575 T^{4} - 161 p T^{5} - 22 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 8 T + 97 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2:C_4$ | \( 1 + 13 T + 67 T^{2} - 115 T^{3} - 3864 T^{4} - 115 p T^{5} + 67 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^2:C_4$ | \( 1 - 6 T + 23 T^{2} - 480 T^{3} + 5581 T^{4} - 480 p T^{5} + 23 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2^2:C_4$ | \( 1 - 15 T + 31 T^{2} + 15 p T^{3} - 10424 T^{4} + 15 p^{2} T^{5} + 31 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2:C_4$ | \( 1 - 3 T + 18 T^{2} + 209 T^{3} + 675 T^{4} + 209 p T^{5} + 18 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 + 4 T + 93 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2:C_4$ | \( 1 + 7 T - 2 T^{2} + 569 T^{3} + 8925 T^{4} + 569 p T^{5} - 2 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2^2:C_4$ | \( 1 + 4 T - 27 T^{2} + 500 T^{3} + 7301 T^{4} + 500 p T^{5} - 27 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2^2:C_4$ | \( 1 - 20 T + 231 T^{2} - 2770 T^{3} + 30671 T^{4} - 2770 p T^{5} + 231 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2^2:C_4$ | \( 1 + 9 T + 53 T^{2} - 75 T^{3} - 4004 T^{4} - 75 p T^{5} + 53 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 15 T + 133 T^{2} + 15 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2:C_4$ | \( 1 - 27 T + 452 T^{2} - 6225 T^{3} + 70951 T^{4} - 6225 p T^{5} + 452 p^{2} T^{6} - 27 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−7.39780419502519003657946752613, −7.21940373644105222411574939089, −6.77127471198367826338498280267, −6.77042695338192185370074880799, −6.68759251267113379360216898246, −6.24325530693751122773790633194, −5.99629316665167882206235074322, −5.94826456117599214476382644125, −5.89640165598970253783798517046, −5.15066802725412647033358178108, −5.07147752182127691440067804132, −4.88635844048059796108448356719, −4.67480165961787760466629956683, −4.17227509825836210736508086389, −3.65206753285261088140367536195, −3.38555689858310900977288386307, −3.22564361907384307049572555752, −3.05683000257495879411149317499, −2.68508729951080178131228737934, −2.44264519778555759286637383380, −2.22073388534095967038950326740, −1.40971799872212725154550220995, −1.19244246522144800736800204536, −0.72019462281082482048584088069, −0.64944023009448949871654120531,
0.64944023009448949871654120531, 0.72019462281082482048584088069, 1.19244246522144800736800204536, 1.40971799872212725154550220995, 2.22073388534095967038950326740, 2.44264519778555759286637383380, 2.68508729951080178131228737934, 3.05683000257495879411149317499, 3.22564361907384307049572555752, 3.38555689858310900977288386307, 3.65206753285261088140367536195, 4.17227509825836210736508086389, 4.67480165961787760466629956683, 4.88635844048059796108448356719, 5.07147752182127691440067804132, 5.15066802725412647033358178108, 5.89640165598970253783798517046, 5.94826456117599214476382644125, 5.99629316665167882206235074322, 6.24325530693751122773790633194, 6.68759251267113379360216898246, 6.77042695338192185370074880799, 6.77127471198367826338498280267, 7.21940373644105222411574939089, 7.39780419502519003657946752613
