| L(s) = 1 | + 2·3-s − 4·5-s − 9-s − 2·11-s + 8·13-s − 8·15-s + 4·17-s − 2·19-s + 4·23-s + 5·25-s − 2·27-s + 12·31-s − 4·33-s + 12·37-s + 16·39-s + 6·43-s + 4·45-s − 18·47-s − 11·49-s + 8·51-s + 10·53-s + 8·55-s − 4·57-s − 6·59-s − 12·61-s − 32·65-s + 22·67-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1.78·5-s − 1/3·9-s − 0.603·11-s + 2.21·13-s − 2.06·15-s + 0.970·17-s − 0.458·19-s + 0.834·23-s + 25-s − 0.384·27-s + 2.15·31-s − 0.696·33-s + 1.97·37-s + 2.56·39-s + 0.914·43-s + 0.596·45-s − 2.62·47-s − 1.57·49-s + 1.12·51-s + 1.37·53-s + 1.07·55-s − 0.529·57-s − 0.781·59-s − 1.53·61-s − 3.96·65-s + 2.68·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.781181989\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.781181989\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| good | 3 | $D_4\times C_2$ | \( 1 - 2 T + 5 T^{2} - 10 T^{3} + 16 T^{4} - 10 p T^{5} + 5 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 2 T - 16 T^{2} - 4 T^{3} + 235 T^{4} - 4 p T^{5} - 16 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 4 T + 20 T^{2} - 100 T^{3} + 271 T^{4} - 100 p T^{5} + 20 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 2 T - 32 T^{2} - 4 T^{3} + 859 T^{4} - 4 p T^{5} - 32 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 4 T + 53 T^{2} - 244 T^{3} + 1588 T^{4} - 244 p T^{5} + 53 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 - 49 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 12 T + 106 T^{2} - 696 T^{3} + 3891 T^{4} - 696 p T^{5} + 106 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2^3$ | \( 1 - 12 T + 72 T^{2} - 288 T^{3} + 983 T^{4} - 288 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 122 T^{2} + 6651 T^{4} - 122 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 6 T + 18 T^{2} - 60 T^{3} - 889 T^{4} - 60 p T^{5} + 18 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 18 T + 90 T^{2} - 528 T^{3} - 8377 T^{4} - 528 p T^{5} + 90 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 14 T + 143 T^{2} - 14 p T^{3} + p^{2} T^{4} )( 1 + 4 T - 37 T^{2} + 4 p T^{3} + p^{2} T^{4} ) \) |
| 59 | $D_4\times C_2$ | \( 1 + 6 T - 64 T^{2} - 108 T^{3} + 4395 T^{4} - 108 p T^{5} - 64 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 12 T + 157 T^{2} + 1308 T^{3} + 11088 T^{4} + 1308 p T^{5} + 157 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 22 T + 137 T^{2} + 834 T^{3} - 16648 T^{4} + 834 p T^{5} + 137 p^{2} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 6 T + 148 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 24 T + 144 T^{2} - 24 p T^{3} - 31057 T^{4} - 24 p^{2} T^{5} + 144 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 6 T + 148 T^{2} + 816 T^{3} + 13203 T^{4} + 816 p T^{5} + 148 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 2 T + 2 T^{2} - 140 T^{3} + 9631 T^{4} - 140 p T^{5} + 2 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2$$\times$$C_2^2$ | \( ( 1 + 16 T + p T^{2} )^{2}( 1 - 16 T + 167 T^{2} - 16 p T^{3} + p^{2} T^{4} ) \) |
| 97 | $D_4\times C_2$ | \( 1 + 4 T + 8 T^{2} + 12 T^{3} - 8818 T^{4} + 12 p T^{5} + 8 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−7.980021468989728217613862527843, −7.76426792423739701654450713441, −7.24811577245422869171332855600, −7.15527135632301220956647719078, −7.08132298194519879940526995035, −6.41338532177598580278298978815, −6.37408812913633825304646863612, −6.06045634547949634619924648105, −5.95722975110971380557796720793, −5.66895033600833930747938904689, −5.30524412108331214043266418814, −4.79477668246437205140889889478, −4.70750490172076158066300270099, −4.42770829381474179927775566291, −4.17277695570369197425993102220, −3.91006254439122032185517464871, −3.57363437185351871256187026249, −3.11902273669145309725029451680, −3.08586664357771231855932281104, −2.92532094509204936491442098845, −2.71980241819848399931898225184, −1.80159770493756240837835762084, −1.73556471878754538975600940548, −0.919768922306661910640116325200, −0.61201967691215579748283958416,
0.61201967691215579748283958416, 0.919768922306661910640116325200, 1.73556471878754538975600940548, 1.80159770493756240837835762084, 2.71980241819848399931898225184, 2.92532094509204936491442098845, 3.08586664357771231855932281104, 3.11902273669145309725029451680, 3.57363437185351871256187026249, 3.91006254439122032185517464871, 4.17277695570369197425993102220, 4.42770829381474179927775566291, 4.70750490172076158066300270099, 4.79477668246437205140889889478, 5.30524412108331214043266418814, 5.66895033600833930747938904689, 5.95722975110971380557796720793, 6.06045634547949634619924648105, 6.37408812913633825304646863612, 6.41338532177598580278298978815, 7.08132298194519879940526995035, 7.15527135632301220956647719078, 7.24811577245422869171332855600, 7.76426792423739701654450713441, 7.980021468989728217613862527843
