| L(s) = 1 | − 7-s − 3·11-s + 2·13-s − 9·17-s − 4·19-s + 3·23-s + 9·29-s + 2·31-s − 37-s − 9·41-s + 8·43-s − 12·47-s − 9·49-s − 12·53-s + 15·59-s − 61-s + 11·67-s − 12·71-s − 10·73-s + 3·77-s − 7·79-s − 12·83-s + 3·89-s − 2·91-s + 5·97-s − 12·101-s − 7·103-s + ⋯ |
| L(s) = 1 | − 0.377·7-s − 0.904·11-s + 0.554·13-s − 2.18·17-s − 0.917·19-s + 0.625·23-s + 1.67·29-s + 0.359·31-s − 0.164·37-s − 1.40·41-s + 1.21·43-s − 1.75·47-s − 9/7·49-s − 1.64·53-s + 1.95·59-s − 0.128·61-s + 1.34·67-s − 1.42·71-s − 1.17·73-s + 0.341·77-s − 0.787·79-s − 1.31·83-s + 0.317·89-s − 0.209·91-s + 0.507·97-s − 1.19·101-s − 0.689·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2 \wr S_4$ | \( 1 + T + 10 T^{2} + 31 T^{3} + 43 T^{4} + 31 p T^{5} + 10 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 + 3 T + 29 T^{2} + 54 T^{3} + 369 T^{4} + 54 p T^{5} + 29 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 - 2 T + 34 T^{2} - 77 T^{3} + 589 T^{4} - 77 p T^{5} + 34 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 + 9 T + 80 T^{2} + 441 T^{3} + 2115 T^{4} + 441 p T^{5} + 80 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 + 4 T + 49 T^{2} + 148 T^{3} + 1117 T^{4} + 148 p T^{5} + 49 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 - 3 T + p T^{2} + 36 T^{3} + 27 T^{4} + 36 p T^{5} + p^{3} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 - 9 T + 56 T^{2} - 117 T^{3} + 405 T^{4} - 117 p T^{5} + 56 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 - 2 T + 106 T^{2} - 149 T^{3} + 4675 T^{4} - 149 p T^{5} + 106 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 + T + 109 T^{2} + 88 T^{3} + 5425 T^{4} + 88 p T^{5} + 109 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 + 9 T + 119 T^{2} + 702 T^{3} + 6477 T^{4} + 702 p T^{5} + 119 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 - 8 T + 118 T^{2} - 608 T^{3} + 6487 T^{4} - 608 p T^{5} + 118 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 + 12 T + 137 T^{2} + 1116 T^{3} + 7461 T^{4} + 1116 p T^{5} + 137 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 + 12 T + 134 T^{2} + 1287 T^{3} + 11367 T^{4} + 1287 p T^{5} + 134 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 - 15 T + 122 T^{2} - 27 T^{3} - 2637 T^{4} - 27 p T^{5} + 122 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 + T + 169 T^{2} - 128 T^{3} + 12859 T^{4} - 128 p T^{5} + 169 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 - 11 T + 184 T^{2} - 1613 T^{3} + 18535 T^{4} - 1613 p T^{5} + 184 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 + 12 T + 212 T^{2} + 1665 T^{3} + 19293 T^{4} + 1665 p T^{5} + 212 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 + 10 T + 280 T^{2} + 1897 T^{3} + 29707 T^{4} + 1897 p T^{5} + 280 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 + 7 T + p T^{2} + 10 p T^{3} + 15493 T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 + 12 T + 236 T^{2} + 2295 T^{3} + 29097 T^{4} + 2295 p T^{5} + 236 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 - 3 T + 113 T^{2} - 1314 T^{3} + 10185 T^{4} - 1314 p T^{5} + 113 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 - 5 T + 253 T^{2} - 788 T^{3} + 32275 T^{4} - 788 p T^{5} + 253 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−6.00659962994456745640274412602, −5.51905893394614992980570515786, −5.50213475612412875479947386138, −5.27807212402313280759644485340, −5.11377706669690631960152396692, −4.85055958598803162123470712445, −4.83092631409782099866266128582, −4.53454331242376413854372865712, −4.48392696390250586300655114402, −4.08637117173596673499974495343, −4.07147101754811109623014292707, −3.85243794091216828562927775736, −3.68530345613344935566368591001, −3.32308879760705803233717268611, −3.14082754301276300994932653605, −2.92161447944607335644854471243, −2.87173859728001879538183307690, −2.38993299736315608820665067153, −2.37957900186146059922151984335, −2.29563858126659840561593786289, −2.09061695355649388246130869645, −1.36575378024868197790115852836, −1.33351687356680280715900374860, −1.23383991444600777823199029133, −1.13231445516284211671684679332, 0, 0, 0, 0,
1.13231445516284211671684679332, 1.23383991444600777823199029133, 1.33351687356680280715900374860, 1.36575378024868197790115852836, 2.09061695355649388246130869645, 2.29563858126659840561593786289, 2.37957900186146059922151984335, 2.38993299736315608820665067153, 2.87173859728001879538183307690, 2.92161447944607335644854471243, 3.14082754301276300994932653605, 3.32308879760705803233717268611, 3.68530345613344935566368591001, 3.85243794091216828562927775736, 4.07147101754811109623014292707, 4.08637117173596673499974495343, 4.48392696390250586300655114402, 4.53454331242376413854372865712, 4.83092631409782099866266128582, 4.85055958598803162123470712445, 5.11377706669690631960152396692, 5.27807212402313280759644485340, 5.50213475612412875479947386138, 5.51905893394614992980570515786, 6.00659962994456745640274412602
