| L(s) = 1 | + 12·5-s − 12·7-s + 36·11-s − 14·13-s − 48·17-s − 36·23-s + 83·25-s − 12·29-s + 72·31-s − 144·35-s + 20·37-s − 108·43-s + 22·49-s + 144·53-s + 432·55-s − 34·61-s − 168·65-s + 204·67-s + 196·73-s − 432·77-s − 276·79-s − 360·83-s − 576·85-s − 96·89-s + 168·91-s + 4·97-s − 216·101-s + ⋯ |
| L(s) = 1 | + 12/5·5-s − 1.71·7-s + 3.27·11-s − 1.07·13-s − 2.82·17-s − 1.56·23-s + 3.31·25-s − 0.413·29-s + 2.32·31-s − 4.11·35-s + 0.540·37-s − 2.51·43-s + 0.448·49-s + 2.71·53-s + 7.85·55-s − 0.557·61-s − 2.58·65-s + 3.04·67-s + 2.68·73-s − 5.61·77-s − 3.49·79-s − 4.33·83-s − 6.77·85-s − 1.07·89-s + 1.84·91-s + 4/97·97-s − 2.13·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(6.185687085\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.185687085\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 12 T + 61 T^{2} - 396 T^{3} + 2664 T^{4} - 396 p^{2} T^{5} + 61 p^{4} T^{6} - 12 p^{6} T^{7} + p^{8} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 + 12 T + 122 T^{2} + 888 T^{3} + 5427 T^{4} + 888 p^{2} T^{5} + 122 p^{4} T^{6} + 12 p^{6} T^{7} + p^{8} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 36 T + 746 T^{2} - 11304 T^{3} + 136227 T^{4} - 11304 p^{2} T^{5} + 746 p^{4} T^{6} - 36 p^{6} T^{7} + p^{8} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 + 14 T - 83 T^{2} - 826 T^{3} + 16156 T^{4} - 826 p^{2} T^{5} - 83 p^{4} T^{6} + 14 p^{6} T^{7} + p^{8} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 + 24 T + 647 T^{2} + 24 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 + 20 p T^{2} + 253542 T^{4} + 20 p^{5} T^{6} + p^{8} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 36 T + 1274 T^{2} + 30312 T^{3} + 657651 T^{4} + 30312 p^{2} T^{5} + 1274 p^{4} T^{6} + 36 p^{6} T^{7} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 12 T - 1571 T^{2} + 396 T^{3} + 2112840 T^{4} + 396 p^{2} T^{5} - 1571 p^{4} T^{6} + 12 p^{6} T^{7} + p^{8} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 72 T + 3938 T^{2} - 159120 T^{3} + 5621187 T^{4} - 159120 p^{2} T^{5} + 3938 p^{4} T^{6} - 72 p^{6} T^{7} + p^{8} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 - 10 T + 1791 T^{2} - 10 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2^3$ | \( 1 - 1634 T^{2} - 155805 T^{4} - 1634 p^{4} T^{6} + p^{8} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 108 T + 8522 T^{2} + 500472 T^{3} + 25244067 T^{4} + 500472 p^{2} T^{5} + 8522 p^{4} T^{6} + 108 p^{6} T^{7} + p^{8} T^{8} \) |
| 47 | $C_2^3$ | \( 1 + 3842 T^{2} + 9881283 T^{4} + 3842 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 72 T + 6482 T^{2} - 72 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 - 2254 T^{2} - 7036845 T^{4} - 2254 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 34 T - 3875 T^{2} - 81974 T^{3} + 7163644 T^{4} - 81974 p^{2} T^{5} - 3875 p^{4} T^{6} + 34 p^{6} T^{7} + p^{8} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 204 T + 25994 T^{2} - 2472888 T^{3} + 189063171 T^{4} - 2472888 p^{2} T^{5} + 25994 p^{4} T^{6} - 204 p^{6} T^{7} + p^{8} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 + 860 T^{2} - 10717626 T^{4} + 860 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 98 T + 6147 T^{2} - 98 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 + 276 T + 43322 T^{2} + 4948680 T^{3} + 441006291 T^{4} + 4948680 p^{2} T^{5} + 43322 p^{4} T^{6} + 276 p^{6} T^{7} + p^{8} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 + 360 T + 67634 T^{2} + 8796240 T^{3} + 847166835 T^{4} + 8796240 p^{2} T^{5} + 67634 p^{4} T^{6} + 360 p^{6} T^{7} + p^{8} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 48 T + 13535 T^{2} + 48 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 4 T - 17078 T^{2} + 6896 T^{3} + 203525011 T^{4} + 6896 p^{2} T^{5} - 17078 p^{4} T^{6} - 4 p^{6} T^{7} + p^{8} T^{8} \) |
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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
−6.66914229477450133511867789797, −6.35873338728458879034202714505, −6.28858958769860732867544577140, −6.27531273827627478186255859546, −5.86016210186991222371459694303, −5.78312001964587221720131286187, −5.54792682364144540202392924973, −5.11192558466757820091021671624, −4.89791432349116992018765500571, −4.52770628099021374624467503898, −4.51907801981395402693671651886, −4.22091178941365282204159049473, −3.94553053541400414741054366247, −3.61510265138563393718496074242, −3.60857492611213019955540877037, −2.98681287020472895625380579782, −2.89990778009561688946954696529, −2.45270664031163449550774736298, −2.21737105032048343649513058927, −2.21316469497181168890350135943, −1.69655867690917594974033883209, −1.47532774153588379325213904260, −1.22050398174176999797263233443, −0.52239763203135771757146576742, −0.40481578220774226118593191164,
0.40481578220774226118593191164, 0.52239763203135771757146576742, 1.22050398174176999797263233443, 1.47532774153588379325213904260, 1.69655867690917594974033883209, 2.21316469497181168890350135943, 2.21737105032048343649513058927, 2.45270664031163449550774736298, 2.89990778009561688946954696529, 2.98681287020472895625380579782, 3.60857492611213019955540877037, 3.61510265138563393718496074242, 3.94553053541400414741054366247, 4.22091178941365282204159049473, 4.51907801981395402693671651886, 4.52770628099021374624467503898, 4.89791432349116992018765500571, 5.11192558466757820091021671624, 5.54792682364144540202392924973, 5.78312001964587221720131286187, 5.86016210186991222371459694303, 6.27531273827627478186255859546, 6.28858958769860732867544577140, 6.35873338728458879034202714505, 6.66914229477450133511867789797
