| L(s) = 1 | + 4·3-s + 4·5-s + 7-s + 10·9-s − 11-s + 10·13-s + 16·15-s + 12·17-s − 5·19-s + 4·21-s − 23-s + 10·25-s + 20·27-s + 2·29-s + 4·31-s − 4·33-s + 4·35-s + 2·37-s + 40·39-s + 4·41-s + 9·43-s + 40·45-s − 11·49-s + 48·51-s + 13·53-s − 4·55-s − 20·57-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 1.78·5-s + 0.377·7-s + 10/3·9-s − 0.301·11-s + 2.77·13-s + 4.13·15-s + 2.91·17-s − 1.14·19-s + 0.872·21-s − 0.208·23-s + 2·25-s + 3.84·27-s + 0.371·29-s + 0.718·31-s − 0.696·33-s + 0.676·35-s + 0.328·37-s + 6.40·39-s + 0.624·41-s + 1.37·43-s + 5.96·45-s − 1.57·49-s + 6.72·51-s + 1.78·53-s − 0.539·55-s − 2.64·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{4} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{4} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(81.42117311\) |
| \(L(\frac12)\) |
\(\approx\) |
\(81.42117311\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{4} \) |
| 5 | $C_1$ | \( ( 1 - T )^{4} \) |
| 31 | $C_1$ | \( ( 1 - T )^{4} \) |
| good | 7 | $C_2 \wr S_4$ | \( 1 - T + 12 T^{2} + 9 T^{3} + 72 T^{4} + 9 p T^{5} + 12 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2^3:S_4$ | \( 1 + T + 18 T^{2} - p T^{3} + 210 T^{4} - p^{2} T^{5} + 18 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 - 10 T + 80 T^{2} - 404 T^{3} + 1730 T^{4} - 404 p T^{5} + 80 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 - 12 T + 92 T^{2} - 512 T^{3} + 2326 T^{4} - 512 p T^{5} + 92 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 + 5 T + 60 T^{2} + 269 T^{3} + 1590 T^{4} + 269 p T^{5} + 60 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 + T + 70 T^{2} + 13 T^{3} + 2126 T^{4} + 13 p T^{5} + 70 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 - 2 T + 50 T^{2} + 128 T^{3} + 882 T^{4} + 128 p T^{5} + 50 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 - 2 T + 140 T^{2} - 212 T^{3} + 7630 T^{4} - 212 p T^{5} + 140 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 - 4 T + 76 T^{2} - 316 T^{3} + 4918 T^{4} - 316 p T^{5} + 76 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 - 9 T + 102 T^{2} - 749 T^{3} + 6938 T^{4} - 749 p T^{5} + 102 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 + 88 T^{2} - 340 T^{3} + 4022 T^{4} - 340 p T^{5} + 88 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 - 13 T + 100 T^{2} - 907 T^{3} + 8954 T^{4} - 907 p T^{5} + 100 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 - 6 T + 226 T^{2} - 1004 T^{3} + 19790 T^{4} - 1004 p T^{5} + 226 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 + 92 T^{2} - 64 T^{3} + 6550 T^{4} - 64 p T^{5} + 92 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 - 8 T + 144 T^{2} - 1346 T^{3} + 13306 T^{4} - 1346 p T^{5} + 144 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 - 9 T + 96 T^{2} + 707 T^{3} - 4836 T^{4} + 707 p T^{5} + 96 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 - 7 T + 174 T^{2} - 1695 T^{3} + 15380 T^{4} - 1695 p T^{5} + 174 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 + 5 T + 172 T^{2} + 813 T^{3} + 20122 T^{4} + 813 p T^{5} + 172 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 + 2 T + 160 T^{2} + 446 T^{3} + 14350 T^{4} + 446 p T^{5} + 160 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 - 5 T + 162 T^{2} - 275 T^{3} + 11628 T^{4} - 275 p T^{5} + 162 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 - 6 T + 100 T^{2} - 2 T^{3} + 10534 T^{4} - 2 p T^{5} + 100 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−5.75591626289392218328171176312, −5.23017388291093848955631883248, −5.10252659909510693796327093686, −5.05645448441592341807839003765, −5.04595070747169236865994520980, −4.37900322119676979512807374267, −4.26351607044107485803282515170, −4.12181279245973676901105923863, −4.07446507061839358857193077352, −3.79565110362366607696997039944, −3.48231305328226216981294892658, −3.36772761864239975422292942489, −3.36732071585213146349351086345, −2.83920439806493858914134216700, −2.82067606946579750425499107484, −2.65024279640150904276286541015, −2.51835581338761973358941057370, −1.99610688103591346413943849070, −1.93760138329469513593956155020, −1.80580433469136978331845073867, −1.56239787300635885807546147641, −1.17013962046377546367148092619, −1.04305229093114937405595290748, −0.829177864174630384100069462689, −0.64268444903941580451669866472,
0.64268444903941580451669866472, 0.829177864174630384100069462689, 1.04305229093114937405595290748, 1.17013962046377546367148092619, 1.56239787300635885807546147641, 1.80580433469136978331845073867, 1.93760138329469513593956155020, 1.99610688103591346413943849070, 2.51835581338761973358941057370, 2.65024279640150904276286541015, 2.82067606946579750425499107484, 2.83920439806493858914134216700, 3.36732071585213146349351086345, 3.36772761864239975422292942489, 3.48231305328226216981294892658, 3.79565110362366607696997039944, 4.07446507061839358857193077352, 4.12181279245973676901105923863, 4.26351607044107485803282515170, 4.37900322119676979512807374267, 5.04595070747169236865994520980, 5.05645448441592341807839003765, 5.10252659909510693796327093686, 5.23017388291093848955631883248, 5.75591626289392218328171176312
