| L(s) = 1 | − 2·2-s + 10·7-s + 4·8-s + 2·11-s + 6·13-s − 20·14-s − 7·16-s + 4·17-s + 4·19-s − 4·22-s − 4·23-s − 12·26-s + 4·29-s − 12·31-s + 4·32-s − 8·34-s + 12·37-s − 8·38-s + 6·41-s + 18·43-s + 8·46-s − 6·47-s + 40·49-s − 8·53-s + 40·56-s − 8·58-s + 8·59-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3.77·7-s + 1.41·8-s + 0.603·11-s + 1.66·13-s − 5.34·14-s − 7/4·16-s + 0.970·17-s + 0.917·19-s − 0.852·22-s − 0.834·23-s − 2.35·26-s + 0.742·29-s − 2.15·31-s + 0.707·32-s − 1.37·34-s + 1.97·37-s − 1.29·38-s + 0.937·41-s + 2.74·43-s + 1.17·46-s − 0.875·47-s + 40/7·49-s − 1.09·53-s + 5.34·56-s − 1.05·58-s + 1.04·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.285359406\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.285359406\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 17 | $C_1$ | \( ( 1 - T )^{4} \) |
| good | 2 | $C_2 \wr S_4$ | \( 1 + p T + p^{2} T^{2} + p^{2} T^{3} + 7 T^{4} + p^{3} T^{5} + p^{4} T^{6} + p^{4} T^{7} + p^{4} T^{8} \) |
| 7 | $C_2 \wr S_4$ | \( 1 - 10 T + 60 T^{2} - 248 T^{3} + 108 p T^{4} - 248 p T^{5} + 60 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 - 2 T + 30 T^{2} - 40 T^{3} + 420 T^{4} - 40 p T^{5} + 30 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 - 6 T + 24 T^{2} - 42 T^{3} + 122 T^{4} - 42 p T^{5} + 24 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 - 4 T + 44 T^{2} - 148 T^{3} + 1062 T^{4} - 148 p T^{5} + 44 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 + 4 T + 72 T^{2} + 194 T^{3} + 2244 T^{4} + 194 p T^{5} + 72 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 - 4 T + 84 T^{2} - 204 T^{3} + 3110 T^{4} - 204 p T^{5} + 84 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 + 12 T + 118 T^{2} + 926 T^{3} + 5468 T^{4} + 926 p T^{5} + 118 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 - 12 T + 164 T^{2} - 1284 T^{3} + 9382 T^{4} - 1284 p T^{5} + 164 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 - 6 T + 128 T^{2} - 586 T^{3} + 7526 T^{4} - 586 p T^{5} + 128 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 - 18 T + 268 T^{2} - 2410 T^{3} + 19034 T^{4} - 2410 p T^{5} + 268 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 + 6 T + 160 T^{2} + 654 T^{3} + 10458 T^{4} + 654 p T^{5} + 160 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 + 8 T + 108 T^{2} + 536 T^{3} + 5846 T^{4} + 536 p T^{5} + 108 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 61 | $C_2 \wr S_4$ | \( 1 - 6 T + 152 T^{2} - 762 T^{3} + 12982 T^{4} - 762 p T^{5} + 152 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 - 6 T + 216 T^{2} - 990 T^{3} + 19914 T^{4} - 990 p T^{5} + 216 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 - 10 T + 266 T^{2} - 1932 T^{3} + 27932 T^{4} - 1932 p T^{5} + 266 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 - 2 T + 116 T^{2} - 262 T^{3} + 7534 T^{4} - 262 p T^{5} + 116 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 + 12 T + 258 T^{2} + 2274 T^{3} + 27756 T^{4} + 2274 p T^{5} + 258 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 - 14 T + 288 T^{2} - 2582 T^{3} + 33074 T^{4} - 2582 p T^{5} + 288 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 + 24 T + 492 T^{2} + 6276 T^{3} + 71250 T^{4} + 6276 p T^{5} + 492 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 + 4 T + 268 T^{2} + 12 p T^{3} + 35174 T^{4} + 12 p^{2} T^{5} + 268 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−5.95428922166921557667965198803, −5.70293881582890574221269340396, −5.52299978392350651887490576382, −5.51560852133941726224678816391, −5.29801850075884527866384540546, −4.83802423080120728577736826120, −4.73454025936446997380097307906, −4.60152493717837053652715971647, −4.59653013401997855411272305037, −4.08526071330608264600337392824, −4.06971628521023310556607006670, −3.96220667055841616500329258125, −3.63191966119162799910998663584, −3.24750352726085415333350358867, −3.24075103432141547744331575367, −2.61547551428463010141248765546, −2.54585628616056230271217576111, −2.14911974624365017232086182618, −1.87043139778204366058939846959, −1.67172703414707151447265011646, −1.56674734235175434901324004946, −1.23632584403016111742337640791, −1.00794816344445499330068130331, −0.66956078201007422413541844219, −0.56475482299069937707439487916,
0.56475482299069937707439487916, 0.66956078201007422413541844219, 1.00794816344445499330068130331, 1.23632584403016111742337640791, 1.56674734235175434901324004946, 1.67172703414707151447265011646, 1.87043139778204366058939846959, 2.14911974624365017232086182618, 2.54585628616056230271217576111, 2.61547551428463010141248765546, 3.24075103432141547744331575367, 3.24750352726085415333350358867, 3.63191966119162799910998663584, 3.96220667055841616500329258125, 4.06971628521023310556607006670, 4.08526071330608264600337392824, 4.59653013401997855411272305037, 4.60152493717837053652715971647, 4.73454025936446997380097307906, 4.83802423080120728577736826120, 5.29801850075884527866384540546, 5.51560852133941726224678816391, 5.52299978392350651887490576382, 5.70293881582890574221269340396, 5.95428922166921557667965198803
