| L(s) = 1 | + 2·3-s + 6·7-s − 2·9-s − 2·11-s − 14·13-s − 14·17-s + 4·19-s + 12·21-s − 4·23-s − 4·27-s + 4·29-s − 4·33-s − 2·37-s − 28·39-s − 8·41-s + 4·43-s + 2·47-s + 4·49-s − 28·51-s − 4·53-s + 8·57-s − 8·61-s − 12·63-s − 2·67-s − 8·69-s + 24·71-s − 18·73-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 2.26·7-s − 2/3·9-s − 0.603·11-s − 3.88·13-s − 3.39·17-s + 0.917·19-s + 2.61·21-s − 0.834·23-s − 0.769·27-s + 0.742·29-s − 0.696·33-s − 0.328·37-s − 4.48·39-s − 1.24·41-s + 0.609·43-s + 0.291·47-s + 4/7·49-s − 3.92·51-s − 0.549·53-s + 1.05·57-s − 1.02·61-s − 1.51·63-s − 0.244·67-s − 0.963·69-s + 2.84·71-s − 2.10·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{8} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{8} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 23 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 3 | $C_2 \wr S_4$ | \( 1 - 2 T + 2 p T^{2} - 4 p T^{3} + 23 T^{4} - 4 p^{2} T^{5} + 2 p^{3} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $C_2 \wr S_4$ | \( 1 - 6 T + 32 T^{2} - 114 T^{3} + 346 T^{4} - 114 p T^{5} + 32 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 + 2 T + 28 T^{2} + 2 p T^{3} + 346 T^{4} + 2 p^{2} T^{5} + 28 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 + 14 T + 118 T^{2} + 664 T^{3} + 2791 T^{4} + 664 p T^{5} + 118 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 + 14 T + 126 T^{2} + 778 T^{3} + 3726 T^{4} + 778 p T^{5} + 126 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 - 4 T + 70 T^{2} - 200 T^{3} + 1918 T^{4} - 200 p T^{5} + 70 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 - 4 T + 94 T^{2} - 344 T^{3} + 3775 T^{4} - 344 p T^{5} + 94 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 + 50 T^{2} + 256 T^{3} + 1011 T^{4} + 256 p T^{5} + 50 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 + 2 T + 76 T^{2} - 158 T^{3} + 2410 T^{4} - 158 p T^{5} + 76 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 + 8 T + 70 T^{2} + 616 T^{3} + 4863 T^{4} + 616 p T^{5} + 70 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 - 4 T + 54 T^{2} - 80 T^{3} + 2910 T^{4} - 80 p T^{5} + 54 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 - 2 T + 50 T^{2} - 116 T^{3} + 4795 T^{4} - 116 p T^{5} + 50 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 + 4 T + 108 T^{2} + 524 T^{3} + 8022 T^{4} + 524 p T^{5} + 108 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 + 216 T^{2} - 16 T^{3} + 18542 T^{4} - 16 p T^{5} + 216 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 + 8 T + 142 T^{2} + 316 T^{3} + 7126 T^{4} + 316 p T^{5} + 142 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 + 2 T + 260 T^{2} + 390 T^{3} + 25866 T^{4} + 390 p T^{5} + 260 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 - 24 T + 350 T^{2} - 3544 T^{3} + 32183 T^{4} - 3544 p T^{5} + 350 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 + 18 T + 270 T^{2} + 2088 T^{3} + 20423 T^{4} + 2088 p T^{5} + 270 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 + 24 T + 494 T^{2} + 6100 T^{3} + 65598 T^{4} + 6100 p T^{5} + 494 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 + 6 T + 62 T^{2} + 518 T^{3} + 11462 T^{4} + 518 p T^{5} + 62 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 + 8 T + 270 T^{2} + 1764 T^{3} + 34598 T^{4} + 1764 p T^{5} + 270 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 + 34 T + 736 T^{2} + 10514 T^{3} + 119290 T^{4} + 10514 p T^{5} + 736 p^{2} T^{6} + 34 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−5.68003398138372503877148533155, −5.36398334895757315742057052128, −5.34092382542241090721270437874, −5.27246857602298186327747397452, −4.94632553188118217973498448743, −4.69640188196280382955325195221, −4.60927529846458005098523663694, −4.59254676685944530530131877551, −4.56699531362554573517798088234, −4.25795478494188673316527509724, −3.99797599399550975876531048676, −3.80907864579985578468176572792, −3.50896585521808108279141877191, −3.05673984847327994104480725788, −2.96496679422016850964782590569, −2.92335430230300634754465166924, −2.80282361265818399858729410706, −2.33004802615365978329871545745, −2.27950654838217029671438233506, −2.24442570192389634416284308539, −2.08898556556481479702670297756, −1.65528332024576492738356464652, −1.61102352966021180348867251699, −1.13870652760093531517582876509, −1.00363082298408494622837727823, 0, 0, 0, 0,
1.00363082298408494622837727823, 1.13870652760093531517582876509, 1.61102352966021180348867251699, 1.65528332024576492738356464652, 2.08898556556481479702670297756, 2.24442570192389634416284308539, 2.27950654838217029671438233506, 2.33004802615365978329871545745, 2.80282361265818399858729410706, 2.92335430230300634754465166924, 2.96496679422016850964782590569, 3.05673984847327994104480725788, 3.50896585521808108279141877191, 3.80907864579985578468176572792, 3.99797599399550975876531048676, 4.25795478494188673316527509724, 4.56699531362554573517798088234, 4.59254676685944530530131877551, 4.60927529846458005098523663694, 4.69640188196280382955325195221, 4.94632553188118217973498448743, 5.27246857602298186327747397452, 5.34092382542241090721270437874, 5.36398334895757315742057052128, 5.68003398138372503877148533155
