| L(s) = 1 | + 4·3-s − 4·5-s + 10·9-s − 2·13-s − 16·15-s + 2·17-s + 6·19-s − 4·23-s + 10·25-s + 20·27-s + 4·29-s + 6·31-s − 4·37-s − 8·39-s + 8·41-s + 20·43-s − 40·45-s − 6·47-s − 9·49-s + 8·51-s − 4·53-s + 24·57-s + 4·59-s − 4·61-s + 8·65-s + 26·67-s − 16·69-s + ⋯ |
| L(s) = 1 | + 2.30·3-s − 1.78·5-s + 10/3·9-s − 0.554·13-s − 4.13·15-s + 0.485·17-s + 1.37·19-s − 0.834·23-s + 2·25-s + 3.84·27-s + 0.742·29-s + 1.07·31-s − 0.657·37-s − 1.28·39-s + 1.24·41-s + 3.04·43-s − 5.96·45-s − 0.875·47-s − 9/7·49-s + 1.12·51-s − 0.549·53-s + 3.17·57-s + 0.520·59-s − 0.512·61-s + 0.992·65-s + 3.17·67-s − 1.92·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(15.40804666\) |
| \(L(\frac12)\) |
\(\approx\) |
\(15.40804666\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{4} \) |
| 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
−5.89034045235034338674463855734, −5.33762548391113341335991498524, −5.16112458275634437687960224084, −5.12704350304752739576578327657, −5.01761452508680353166807614412, −4.48309361363961771302663599825, −4.47986372596935319762172090626, −4.37567095419334721432429136212, −4.24535345093189590028994466593, −3.83408782567798964223813986683, −3.73496060965209930487038373426, −3.44149420932361365039407454792, −3.38433344288117262657466253226, −3.27996776677865521372040505375, −2.89194634994068628803332016180, −2.84720000562827614113106750669, −2.62251557337604534733895639508, −2.21802730382417712860980062184, −2.04228658210108781783503645722, −2.00121433236133722783160478122, −1.61711253381787129786102791470, −1.07680411317122498738490689733, −0.864257449423868844617687497094, −0.74992601769873561330195762976, −0.44719517368359605850515577495,
0.44719517368359605850515577495, 0.74992601769873561330195762976, 0.864257449423868844617687497094, 1.07680411317122498738490689733, 1.61711253381787129786102791470, 2.00121433236133722783160478122, 2.04228658210108781783503645722, 2.21802730382417712860980062184, 2.62251557337604534733895639508, 2.84720000562827614113106750669, 2.89194634994068628803332016180, 3.27996776677865521372040505375, 3.38433344288117262657466253226, 3.44149420932361365039407454792, 3.73496060965209930487038373426, 3.83408782567798964223813986683, 4.24535345093189590028994466593, 4.37567095419334721432429136212, 4.47986372596935319762172090626, 4.48309361363961771302663599825, 5.01761452508680353166807614412, 5.12704350304752739576578327657, 5.16112458275634437687960224084, 5.33762548391113341335991498524, 5.89034045235034338674463855734
