| L(s) = 1 | + 7-s − 3·9-s + 11-s − 3·13-s − 2·17-s − 15·19-s − 4·23-s − 9·27-s + 29-s − 12·31-s − 18·37-s − 9·41-s + 9·43-s + 4·47-s − 15·49-s − 5·59-s + 14·61-s − 3·63-s + 16·67-s − 2·71-s − 21·73-s + 77-s − 21·79-s + 11·81-s + 9·83-s − 16·89-s − 3·91-s + ⋯ |
| L(s) = 1 | + 0.377·7-s − 9-s + 0.301·11-s − 0.832·13-s − 0.485·17-s − 3.44·19-s − 0.834·23-s − 1.73·27-s + 0.185·29-s − 2.15·31-s − 2.95·37-s − 1.40·41-s + 1.37·43-s + 0.583·47-s − 2.14·49-s − 0.650·59-s + 1.79·61-s − 0.377·63-s + 1.95·67-s − 0.237·71-s − 2.45·73-s + 0.113·77-s − 2.36·79-s + 11/9·81-s + 0.987·83-s − 1.69·89-s − 0.314·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{8} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 23 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 3 | $C_2 \wr S_4$ | \( 1 + p T^{2} + p^{2} T^{3} - 2 T^{4} + p^{3} T^{5} + p^{3} T^{6} + p^{4} T^{8} \) |
| 7 | $C_2 \wr S_4$ | \( 1 - T + 16 T^{2} - 25 T^{3} + 134 T^{4} - 25 p T^{5} + 16 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 - T + 2 p T^{2} - 49 T^{3} + 282 T^{4} - 49 p T^{5} + 2 p^{3} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 + 3 T + 21 T^{2} - 33 T^{3} + 41 T^{4} - 33 p T^{5} + 21 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 + 2 T + 44 T^{2} + 110 T^{3} + 950 T^{4} + 110 p T^{5} + 44 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 + 15 T + 138 T^{2} + 871 T^{3} + 4338 T^{4} + 871 p T^{5} + 138 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 - T + 45 T^{2} - 9 T^{3} + 1821 T^{4} - 9 p T^{5} + 45 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 + 12 T + 145 T^{2} + 1035 T^{3} + 7114 T^{4} + 1035 p T^{5} + 145 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 + 18 T + 216 T^{2} + 1854 T^{3} + 12494 T^{4} + 1854 p T^{5} + 216 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 + 9 T + 181 T^{2} + 1089 T^{3} + 11433 T^{4} + 1089 p T^{5} + 181 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 - 9 T + 132 T^{2} - 905 T^{3} + 8166 T^{4} - 905 p T^{5} + 132 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 - 4 T + 181 T^{2} - 541 T^{3} + 12612 T^{4} - 541 p T^{5} + 181 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 + 68 T^{2} - 576 T^{3} + 1078 T^{4} - 576 p T^{5} + 68 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 + 5 T + 61 T^{2} + 672 T^{3} + 6482 T^{4} + 672 p T^{5} + 61 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 - 14 T + 216 T^{2} - 1602 T^{3} + 16174 T^{4} - 1602 p T^{5} + 216 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 - 16 T + 328 T^{2} - 3256 T^{3} + 34910 T^{4} - 3256 p T^{5} + 328 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 + 2 T + 225 T^{2} + 357 T^{3} + 22390 T^{4} + 357 p T^{5} + 225 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 + 21 T + 397 T^{2} + 4569 T^{3} + 46957 T^{4} + 4569 p T^{5} + 397 p^{2} T^{6} + 21 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 + 21 T + 208 T^{2} + 2577 T^{3} + 30046 T^{4} + 2577 p T^{5} + 208 p^{2} T^{6} + 21 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 - 9 T + 216 T^{2} - 1629 T^{3} + 26326 T^{4} - 1629 p T^{5} + 216 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 + 16 T + 400 T^{2} + 4120 T^{3} + 55278 T^{4} + 4120 p T^{5} + 400 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 + 40 T + 944 T^{2} + 14688 T^{3} + 169342 T^{4} + 14688 p T^{5} + 944 p^{2} T^{6} + 40 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−6.45834076435318028309517493937, −5.86743619261854274300875982819, −5.80790045329829160782910478281, −5.65060302673315339281015554682, −5.55709556741293989735054524575, −5.29547466819760967664368326168, −5.19934426504816359349779588065, −4.86110979828594436307384393466, −4.80396264292734871010797740899, −4.29355689803287847164499488427, −4.18663731156880236898553371349, −4.10271561824175222117325266925, −4.09916258794734414136933053470, −3.67337974666068965426265398728, −3.45432655395725544827937423637, −3.32604502721366071656954252932, −3.02844950842706846661092428338, −2.68747899364233788582371986636, −2.40887977318521730860130601499, −2.22563877464118987544218866657, −2.20183238143414065373927022110, −1.70609909277831241945519465781, −1.64040302388613705648717578181, −1.55511679369242570502782718082, −0.973656284590585275174486840505, 0, 0, 0, 0,
0.973656284590585275174486840505, 1.55511679369242570502782718082, 1.64040302388613705648717578181, 1.70609909277831241945519465781, 2.20183238143414065373927022110, 2.22563877464118987544218866657, 2.40887977318521730860130601499, 2.68747899364233788582371986636, 3.02844950842706846661092428338, 3.32604502721366071656954252932, 3.45432655395725544827937423637, 3.67337974666068965426265398728, 4.09916258794734414136933053470, 4.10271561824175222117325266925, 4.18663731156880236898553371349, 4.29355689803287847164499488427, 4.80396264292734871010797740899, 4.86110979828594436307384393466, 5.19934426504816359349779588065, 5.29547466819760967664368326168, 5.55709556741293989735054524575, 5.65060302673315339281015554682, 5.80790045329829160782910478281, 5.86743619261854274300875982819, 6.45834076435318028309517493937
