| L(s) = 1 | + 4·5-s − 2·13-s − 2·17-s − 6·19-s − 4·23-s + 10·25-s − 4·29-s − 6·31-s − 4·37-s − 8·41-s − 20·43-s − 6·47-s − 9·49-s + 4·53-s + 4·59-s − 4·61-s − 8·65-s − 26·67-s − 18·73-s − 18·79-s + 26·83-s − 8·85-s − 14·89-s − 24·95-s − 12·97-s − 12·101-s − 38·103-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 0.554·13-s − 0.485·17-s − 1.37·19-s − 0.834·23-s + 2·25-s − 0.742·29-s − 1.07·31-s − 0.657·37-s − 1.24·41-s − 3.04·43-s − 0.875·47-s − 9/7·49-s + 0.549·53-s + 0.520·59-s − 0.512·61-s − 0.992·65-s − 3.17·67-s − 2.10·73-s − 2.02·79-s + 2.85·83-s − 0.867·85-s − 1.48·89-s − 2.46·95-s − 1.21·97-s − 1.19·101-s − 3.74·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 5^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 5^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{4} \) |
| 23 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 7 | $C_2 \wr S_4$ | \( 1 + 9 T^{2} - 6 T^{3} + 36 T^{4} - 6 p T^{5} + 9 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 + 2 p T^{2} - 12 T^{3} + 322 T^{4} - 12 p T^{5} + 2 p^{3} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 + 2 T + 14 T^{2} + 22 T^{3} + 322 T^{4} + 22 p T^{5} + 14 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 + 2 T - 3 T^{2} + 44 T^{3} + 344 T^{4} + 44 p T^{5} - 3 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 + 6 T + 50 T^{2} + 230 T^{3} + 1434 T^{4} + 230 p T^{5} + 50 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 + 4 T + 41 T^{2} - 52 T^{3} + 532 T^{4} - 52 p T^{5} + 41 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 + 6 T + 113 T^{2} + 454 T^{3} + 4956 T^{4} + 454 p T^{5} + 113 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 + 4 T + 53 T^{2} + 298 T^{3} + 3288 T^{4} + 298 p T^{5} + 53 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 + 8 T + 153 T^{2} + 780 T^{3} + 8804 T^{4} + 780 p T^{5} + 153 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 + 20 T + 240 T^{2} + 2116 T^{3} + 14894 T^{4} + 2116 p T^{5} + 240 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 + 6 T + 162 T^{2} + 734 T^{3} + 11066 T^{4} + 734 p T^{5} + 162 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 - 4 T + 65 T^{2} - 812 T^{3} + 2740 T^{4} - 812 p T^{5} + 65 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 - 4 T + 161 T^{2} - 792 T^{3} + 12356 T^{4} - 792 p T^{5} + 161 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 + 4 T + 182 T^{2} + 368 T^{3} + 14362 T^{4} + 368 p T^{5} + 182 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 + 26 T + 409 T^{2} + 4326 T^{3} + 38852 T^{4} + 4326 p T^{5} + 409 p^{2} T^{6} + 26 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 + 249 T^{2} - 96 T^{3} + 25212 T^{4} - 96 p T^{5} + 249 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 + 18 T + 374 T^{2} + 3934 T^{3} + 43794 T^{4} + 3934 p T^{5} + 374 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 + 18 T + 260 T^{2} + 2730 T^{3} + 29110 T^{4} + 2730 p T^{5} + 260 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 - 26 T + 441 T^{2} - 4988 T^{3} + 50252 T^{4} - 4988 p T^{5} + 441 p^{2} T^{6} - 26 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 + 14 T + 380 T^{2} + 3690 T^{3} + 51734 T^{4} + 3690 p T^{5} + 380 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 + 12 T + 344 T^{2} + 3364 T^{3} + 47982 T^{4} + 3364 p T^{5} + 344 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−5.86594979083031180282332150257, −5.70041822655607747383645254571, −5.58516148744577378532974620790, −5.18223676055848971802500436494, −5.14089533164383053590653172570, −4.94835707725241661566473900987, −4.77735924276445835718726252666, −4.72097132955735580579562722873, −4.45375073071408578638415969709, −4.03735624717594041291722108936, −4.01395501655955107898909581193, −3.91890786403089708644430837746, −3.56263185134676744444592039463, −3.23467766929666216753724479079, −3.22892147656683882581059240387, −2.91821318284787962862050081776, −2.85817745118355872611480105466, −2.35156687073929550709998312447, −2.31283154840488540545568045519, −2.08902931574829899628105974442, −1.98113897031378407103206719384, −1.50350791414627311153038069234, −1.47276135087689712586132679436, −1.32929864836062096843904330347, −1.18602365354838743363751289778, 0, 0, 0, 0,
1.18602365354838743363751289778, 1.32929864836062096843904330347, 1.47276135087689712586132679436, 1.50350791414627311153038069234, 1.98113897031378407103206719384, 2.08902931574829899628105974442, 2.31283154840488540545568045519, 2.35156687073929550709998312447, 2.85817745118355872611480105466, 2.91821318284787962862050081776, 3.22892147656683882581059240387, 3.23467766929666216753724479079, 3.56263185134676744444592039463, 3.91890786403089708644430837746, 4.01395501655955107898909581193, 4.03735624717594041291722108936, 4.45375073071408578638415969709, 4.72097132955735580579562722873, 4.77735924276445835718726252666, 4.94835707725241661566473900987, 5.14089533164383053590653172570, 5.18223676055848971802500436494, 5.58516148744577378532974620790, 5.70041822655607747383645254571, 5.86594979083031180282332150257
