| L(s) = 1 | + 4·2-s − 3-s + 10·4-s + 5-s − 4·6-s + 20·8-s − 9-s + 4·10-s + 7·11-s − 10·12-s + 7·13-s − 15-s + 35·16-s − 4·18-s − 2·19-s + 10·20-s + 28·22-s + 6·23-s − 20·24-s − 5·25-s + 28·26-s − 2·27-s + 4·29-s − 4·30-s + 7·31-s + 56·32-s − 7·33-s + ⋯ |
| L(s) = 1 | + 2.82·2-s − 0.577·3-s + 5·4-s + 0.447·5-s − 1.63·6-s + 7.07·8-s − 1/3·9-s + 1.26·10-s + 2.11·11-s − 2.88·12-s + 1.94·13-s − 0.258·15-s + 35/4·16-s − 0.942·18-s − 0.458·19-s + 2.23·20-s + 5.96·22-s + 1.25·23-s − 4.08·24-s − 25-s + 5.49·26-s − 0.384·27-s + 0.742·29-s − 0.730·30-s + 1.25·31-s + 9.89·32-s − 1.21·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 7^{8} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 7^{8} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(68.30451848\) |
| \(L(\frac12)\) |
\(\approx\) |
\(68.30451848\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{4} \) |
| 7 | | \( 1 \) |
| 29 | $C_1$ | \( ( 1 - T )^{4} \) |
| good | 3 | $C_2^3:S_4$ | \( 1 + T + 2 T^{2} + 5 T^{3} + 2 T^{4} + 5 p T^{5} + 2 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 5 | $C_2 \wr S_4$ | \( 1 - T + 6 T^{2} + 9 T^{3} + 14 T^{4} + 9 p T^{5} + 6 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 - 7 T + 52 T^{2} - 215 T^{3} + 886 T^{4} - 215 p T^{5} + 52 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 - 7 T + 2 p T^{2} - 97 T^{3} + 366 T^{4} - 97 p T^{5} + 2 p^{3} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 + 48 T^{2} + 36 T^{3} + 1038 T^{4} + 36 p T^{5} + 48 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 + 2 T + 56 T^{2} + 106 T^{3} + 1438 T^{4} + 106 p T^{5} + 56 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 - 6 T + 84 T^{2} - 350 T^{3} + 2774 T^{4} - 350 p T^{5} + 84 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 - 7 T + 2 p T^{2} - 263 T^{3} + 2278 T^{4} - 263 p T^{5} + 2 p^{3} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 + 12 T + 172 T^{2} + 1268 T^{3} + 9974 T^{4} + 1268 p T^{5} + 172 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 + 4 T + 16 T^{2} - 280 T^{3} - 2498 T^{4} - 280 p T^{5} + 16 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 + 5 T + 140 T^{2} + 581 T^{3} + 8598 T^{4} + 581 p T^{5} + 140 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 - 11 T + 178 T^{2} - 1315 T^{3} + 258 p T^{4} - 1315 p T^{5} + 178 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 - 5 T + 166 T^{2} - 535 T^{3} + 11730 T^{4} - 535 p T^{5} + 166 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 + 16 T + 220 T^{2} + 1916 T^{3} + 16246 T^{4} + 1916 p T^{5} + 220 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 - 34 T + 612 T^{2} - 7358 T^{3} + 65846 T^{4} - 7358 p T^{5} + 612 p^{2} T^{6} - 34 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 - 2 T - 52 T^{2} - 146 T^{3} + 8886 T^{4} - 146 p T^{5} - 52 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 - 24 T + 228 T^{2} - 328 T^{3} - 6250 T^{4} - 328 p T^{5} + 228 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 - 24 T + 424 T^{2} - 5164 T^{3} + 51182 T^{4} - 5164 p T^{5} + 424 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 + 9 T + 128 T^{2} + 1221 T^{3} + 7198 T^{4} + 1221 p T^{5} + 128 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 - 8 T + 36 T^{2} - 484 T^{3} + 9222 T^{4} - 484 p T^{5} + 36 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 + 2 T + 232 T^{2} + 202 T^{3} + 26278 T^{4} + 202 p T^{5} + 232 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 - 4 T + 240 T^{2} - 392 T^{3} + 27294 T^{4} - 392 p T^{5} + 240 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−6.37741476250139304006047227522, −5.82744255102000006438462434450, −5.72052399413587787706032180088, −5.64013647306305381894656928131, −5.37894885431950560107408604274, −5.20094427155168694073552820599, −5.19721507174371057103906874633, −4.69382799511575644186469000473, −4.55979528742455079090907860406, −4.38231729599085815553609178658, −4.16237172599064618874169615196, −3.84462604307162833104251526975, −3.78096987423519483742475532066, −3.57887725069337121021571424058, −3.31468629244056165646310472450, −3.19319162996491952128113289329, −3.12969298025938860455808253696, −2.40813150379574758101706904241, −2.33123322764234716007824945573, −2.04141557686922445144436253392, −1.78380702996378436573054838110, −1.74442185043921546433729771343, −0.998895007159384917109120404753, −0.852481726925431566562754685540, −0.804365841836234152321066814471,
0.804365841836234152321066814471, 0.852481726925431566562754685540, 0.998895007159384917109120404753, 1.74442185043921546433729771343, 1.78380702996378436573054838110, 2.04141557686922445144436253392, 2.33123322764234716007824945573, 2.40813150379574758101706904241, 3.12969298025938860455808253696, 3.19319162996491952128113289329, 3.31468629244056165646310472450, 3.57887725069337121021571424058, 3.78096987423519483742475532066, 3.84462604307162833104251526975, 4.16237172599064618874169615196, 4.38231729599085815553609178658, 4.55979528742455079090907860406, 4.69382799511575644186469000473, 5.19721507174371057103906874633, 5.20094427155168694073552820599, 5.37894885431950560107408604274, 5.64013647306305381894656928131, 5.72052399413587787706032180088, 5.82744255102000006438462434450, 6.37741476250139304006047227522
