| L(s) = 1 | − 2-s − 4-s + 2·5-s + 3·8-s − 2·10-s − 5·11-s + 5·13-s − 4·16-s + 6·17-s + 8·19-s − 2·20-s + 5·22-s + 12·23-s − 4·25-s − 5·26-s + 10·29-s + 18·31-s + 32-s − 6·34-s − 8·38-s + 6·40-s − 5·41-s − 7·43-s + 5·44-s − 12·46-s − 21·47-s + 4·50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.894·5-s + 1.06·8-s − 0.632·10-s − 1.50·11-s + 1.38·13-s − 16-s + 1.45·17-s + 1.83·19-s − 0.447·20-s + 1.06·22-s + 2.50·23-s − 4/5·25-s − 0.980·26-s + 1.85·29-s + 3.23·31-s + 0.176·32-s − 1.02·34-s − 1.29·38-s + 0.948·40-s − 0.780·41-s − 1.06·43-s + 0.753·44-s − 1.76·46-s − 3.06·47-s + 0.565·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.043533906\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.043533906\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2 \wr S_4$ | \( 1 + T + p T^{2} + 3 T^{4} + p^{3} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 5 | $C_2 \wr S_4$ | \( 1 - 2 T + 8 T^{2} + 3 T^{3} + 9 T^{4} + 3 p T^{5} + 8 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 + 5 T + 20 T^{2} - 9 T^{3} - 51 T^{4} - 9 p T^{5} + 20 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 - 5 T + 4 p T^{2} - 175 T^{3} + 1007 T^{4} - 175 p T^{5} + 4 p^{3} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 - 6 T + 23 T^{2} + 48 T^{3} - 363 T^{4} + 48 p T^{5} + 23 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 - 8 T + 4 p T^{2} - 409 T^{3} + 2117 T^{4} - 409 p T^{5} + 4 p^{3} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 - 12 T + 128 T^{2} - 861 T^{3} + 4839 T^{4} - 861 p T^{5} + 128 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 - 10 T + 4 p T^{2} - 780 T^{3} + 5109 T^{4} - 780 p T^{5} + 4 p^{3} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 - 18 T + 7 p T^{2} - 1810 T^{3} + 11511 T^{4} - 1810 p T^{5} + 7 p^{3} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 + 70 T^{2} + p T^{3} + 3393 T^{4} + p^{2} T^{5} + 70 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 + 5 T + 92 T^{2} + 489 T^{3} + 4623 T^{4} + 489 p T^{5} + 92 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 + 7 T + 142 T^{2} + 727 T^{3} + 8563 T^{4} + 727 p T^{5} + 142 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 + 21 T + 341 T^{2} + 3408 T^{3} + 28077 T^{4} + 3408 p T^{5} + 341 p^{2} T^{6} + 21 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 - 12 T + 4 p T^{2} - 1803 T^{3} + 16773 T^{4} - 1803 p T^{5} + 4 p^{3} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 - 6 T + 128 T^{2} - 861 T^{3} + 8331 T^{4} - 861 p T^{5} + 128 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 - 20 T + 271 T^{2} - 3010 T^{3} + 26663 T^{4} - 3010 p T^{5} + 271 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 + 5 T + 196 T^{2} + 931 T^{3} + 17639 T^{4} + 931 p T^{5} + 196 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 - 9 T + 257 T^{2} - 1782 T^{3} + 26655 T^{4} - 1782 p T^{5} + 257 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 - 6 T + 142 T^{2} - 19 p T^{3} + 12363 T^{4} - 19 p^{2} T^{5} + 142 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 + 10 T + 232 T^{2} + 2365 T^{3} + 24181 T^{4} + 2365 p T^{5} + 232 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 - 9 T + 206 T^{2} - 531 T^{3} + 15315 T^{4} - 531 p T^{5} + 206 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 + 22 T + 470 T^{2} + 5925 T^{3} + 67797 T^{4} + 5925 p T^{5} + 470 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 - 9 T + 130 T^{2} - 163 T^{3} + 9279 T^{4} - 163 p T^{5} + 130 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−6.01055280350625975904708096597, −5.57563035463830511352757176983, −5.50869899506020740708633454349, −5.40490235577959264150626994386, −5.27581488205400168159000184543, −4.82382093248910994549807705560, −4.82041202547137967204795739007, −4.79919316221253403618658301115, −4.68068539423066181902152811145, −4.04451460267217978763443351780, −3.84622118287558747075970123859, −3.79071963029528475234662684786, −3.57820455095847376851193401557, −3.13718450495830824514778001766, −2.94325278308009149145132527387, −2.85936064035666595500632604063, −2.69671159140971554176645505054, −2.40813954636209964861114108291, −2.19445589651888862089944549768, −1.52785505096270780706779402802, −1.43650366893265572994786448288, −1.21790938558758227191926983072, −1.11082190766082195375261712101, −0.62039039053438731812807242327, −0.44084072655410778104280363230,
0.44084072655410778104280363230, 0.62039039053438731812807242327, 1.11082190766082195375261712101, 1.21790938558758227191926983072, 1.43650366893265572994786448288, 1.52785505096270780706779402802, 2.19445589651888862089944549768, 2.40813954636209964861114108291, 2.69671159140971554176645505054, 2.85936064035666595500632604063, 2.94325278308009149145132527387, 3.13718450495830824514778001766, 3.57820455095847376851193401557, 3.79071963029528475234662684786, 3.84622118287558747075970123859, 4.04451460267217978763443351780, 4.68068539423066181902152811145, 4.79919316221253403618658301115, 4.82041202547137967204795739007, 4.82382093248910994549807705560, 5.27581488205400168159000184543, 5.40490235577959264150626994386, 5.50869899506020740708633454349, 5.57563035463830511352757176983, 6.01055280350625975904708096597
