L(s) = 1 | + 2·5-s + 12·11-s + 6·13-s + 4·17-s + 8·19-s − 2·23-s + 6·25-s − 8·29-s − 12·37-s − 12·41-s − 8·43-s + 4·47-s − 18·49-s + 6·53-s + 24·55-s + 2·59-s + 6·61-s + 12·65-s − 12·67-s + 12·71-s − 22·73-s − 8·79-s − 40·83-s + 8·85-s − 6·89-s + 16·95-s − 12·97-s + ⋯ |
L(s) = 1 | + 0.894·5-s + 3.61·11-s + 1.66·13-s + 0.970·17-s + 1.83·19-s − 0.417·23-s + 6/5·25-s − 1.48·29-s − 1.97·37-s − 1.87·41-s − 1.21·43-s + 0.583·47-s − 2.57·49-s + 0.824·53-s + 3.23·55-s + 0.260·59-s + 0.768·61-s + 1.48·65-s − 1.46·67-s + 1.42·71-s − 2.57·73-s − 0.900·79-s − 4.39·83-s + 0.867·85-s − 0.635·89-s + 1.64·95-s − 1.21·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4502406883\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4502406883\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
good | 5 | $D_4\times C_2$ | \( 1 - 2 T - 2 T^{2} + 8 T^{3} - 9 T^{4} + 8 p T^{5} - 2 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $C_2^2$ | \( ( 1 + 9 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_4$ | \( ( 1 - 6 T + 26 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 3 T - 4 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 4 T - 2 T^{2} + 64 T^{3} - 237 T^{4} + 64 p T^{5} - 2 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2^3$ | \( 1 + 2 T + 2 T^{2} - 88 T^{3} - 617 T^{4} - 88 p T^{5} + 2 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2^2$ | \( ( 1 + 4 T - 13 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + 57 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 + 6 T + 63 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 + 12 T + 46 T^{2} + 192 T^{3} + 2019 T^{4} + 192 p T^{5} + 46 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 8 T + 7 T^{2} - 232 T^{3} - 1352 T^{4} - 232 p T^{5} + 7 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 4 T - 2 T^{2} + 304 T^{3} - 2637 T^{4} + 304 p T^{5} - 2 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 6 T - 34 T^{2} + 216 T^{3} + 183 T^{4} + 216 p T^{5} - 34 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 2 T - 70 T^{2} + 88 T^{3} + 1759 T^{4} + 88 p T^{5} - 70 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 6 T - 75 T^{2} + 66 T^{3} + 6404 T^{4} + 66 p T^{5} - 75 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 12 T + 19 T^{2} - 108 T^{3} + 1488 T^{4} - 108 p T^{5} + 19 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 - 6 T - 35 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 22 T + 237 T^{2} + 2222 T^{3} + 20348 T^{4} + 2222 p T^{5} + 237 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 8 T - 105 T^{2} + 88 T^{3} + 17264 T^{4} + 88 p T^{5} - 105 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 20 T + 246 T^{2} + 20 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 6 T - 146 T^{2} + 24 T^{3} + 21999 T^{4} + 24 p T^{5} - 146 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2^2$ | \( ( 1 + 6 T - 61 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−6.34678415233187133064411436820, −6.09669187746509661369896612740, −5.84060104536606142856221669436, −5.51874225723015156309468989094, −5.44712800029526734493210183470, −5.34684211785002010507471444208, −5.28861929779042955106746044336, −4.87818185501595166568416759774, −4.46163109045909818502266688824, −4.23735335990814421189243268080, −4.19176418920546634469511351228, −3.82481163729046829257737886004, −3.71115985798994371414353130355, −3.50744271668814727095435110732, −3.44438683856077930559899089529, −2.98410860140377000452933656512, −2.79603935292961759284962409606, −2.74677771665753045878679494247, −2.08773622048147127925910861690, −1.60800656598601025234192325256, −1.47811162644467992562537684705, −1.34134667238122383096981230270, −1.33668146358957653154370131508, −1.20076775961836824117367246072, −0.07855784519228937347203966286,
0.07855784519228937347203966286, 1.20076775961836824117367246072, 1.33668146358957653154370131508, 1.34134667238122383096981230270, 1.47811162644467992562537684705, 1.60800656598601025234192325256, 2.08773622048147127925910861690, 2.74677771665753045878679494247, 2.79603935292961759284962409606, 2.98410860140377000452933656512, 3.44438683856077930559899089529, 3.50744271668814727095435110732, 3.71115985798994371414353130355, 3.82481163729046829257737886004, 4.19176418920546634469511351228, 4.23735335990814421189243268080, 4.46163109045909818502266688824, 4.87818185501595166568416759774, 5.28861929779042955106746044336, 5.34684211785002010507471444208, 5.44712800029526734493210183470, 5.51874225723015156309468989094, 5.84060104536606142856221669436, 6.09669187746509661369896612740, 6.34678415233187133064411436820