| L(s) = 1 | + 4·7-s + 6·13-s − 12·17-s − 16·19-s − 8·25-s − 2·43-s − 12·47-s − 6·49-s + 24·59-s + 2·61-s + 6·67-s + 12·71-s + 14·73-s − 18·79-s + 24·91-s − 36·97-s + 36·101-s + 24·107-s − 48·119-s + 4·121-s + 127-s + 131-s − 64·133-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 1.51·7-s + 1.66·13-s − 2.91·17-s − 3.67·19-s − 8/5·25-s − 0.304·43-s − 1.75·47-s − 6/7·49-s + 3.12·59-s + 0.256·61-s + 0.733·67-s + 1.42·71-s + 1.63·73-s − 2.02·79-s + 2.51·91-s − 3.65·97-s + 3.58·101-s + 2.32·107-s − 4.40·119-s + 4/11·121-s + 0.0887·127-s + 0.0873·131-s − 5.54·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-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
| \(L(1)\) |
\(\approx\) |
\(0.05846893537\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.05846893537\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| good | 5 | $C_2^3$ | \( 1 + 8 T^{2} + 39 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | $D_{4}$ | \( ( 1 - 2 T + 9 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 4 T^{2} - 138 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 6 T + 23 T^{2} - 66 T^{3} + 108 T^{4} - 66 p T^{5} + 23 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 12 T + 92 T^{2} + 528 T^{3} + 2463 T^{4} + 528 p T^{5} + 92 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2^3$ | \( 1 + 14 T^{2} - 333 T^{4} + 14 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2^3$ | \( 1 - 52 T^{2} + 1863 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 34 T^{2} + 267 T^{4} - 34 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 106 T^{2} + 5331 T^{4} - 106 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 + 2 T - 59 T^{2} - 46 T^{3} + 1948 T^{4} - 46 p T^{5} - 59 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 12 T + 152 T^{2} + 1248 T^{3} + 10863 T^{4} + 1248 p T^{5} + 152 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^3$ | \( 1 - 82 T^{2} + 3915 T^{4} - 82 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 24 T + 320 T^{2} - 3312 T^{3} + 28071 T^{4} - 3312 p T^{5} + 320 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 2 T - 113 T^{2} + 10 T^{3} + 9724 T^{4} + 10 p T^{5} - 113 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 6 T + 77 T^{2} - 390 T^{3} + 540 T^{4} - 390 p T^{5} + 77 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 - 6 T - 35 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 14 T + 25 T^{2} - 350 T^{3} + 9604 T^{4} - 350 p T^{5} + 25 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 18 T + 275 T^{2} + 3006 T^{3} + 30180 T^{4} + 3006 p T^{5} + 275 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 112 T^{2} + 16050 T^{4} - 112 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2^3$ | \( 1 - 28 T^{2} - 7137 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 36 T + 662 T^{2} + 8280 T^{3} + 85395 T^{4} + 8280 p T^{5} + 662 p^{2} T^{6} + 36 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−6.28498501606813475465846017659, −6.10298626515778721430123935003, −5.98157773853484316821630077618, −5.81034345443413883564674616634, −5.28931250779234475083679541220, −5.14350651948415531947143695755, −4.98239696342815774093503302745, −4.81860962682089023131791456127, −4.57413233666992765343874673778, −4.39480181769962949658977673957, −4.05461130635036470668516310782, −4.05252640568391062345891931900, −3.83445524858426397345178470961, −3.62940711054893405007587423713, −3.41127217434658786234463635615, −2.95415841897493586841199976714, −2.60255352649336640045602417744, −2.26964039749189624848372912621, −2.12774421401473185936221484510, −2.05283034014670109612566851262, −1.87481873176743822911907625510, −1.46532100779203454900093783724, −1.19262419729026175686604415984, −0.61720684680204542347803852025, −0.04183068726093175586842652065,
0.04183068726093175586842652065, 0.61720684680204542347803852025, 1.19262419729026175686604415984, 1.46532100779203454900093783724, 1.87481873176743822911907625510, 2.05283034014670109612566851262, 2.12774421401473185936221484510, 2.26964039749189624848372912621, 2.60255352649336640045602417744, 2.95415841897493586841199976714, 3.41127217434658786234463635615, 3.62940711054893405007587423713, 3.83445524858426397345178470961, 4.05252640568391062345891931900, 4.05461130635036470668516310782, 4.39480181769962949658977673957, 4.57413233666992765343874673778, 4.81860962682089023131791456127, 4.98239696342815774093503302745, 5.14350651948415531947143695755, 5.28931250779234475083679541220, 5.81034345443413883564674616634, 5.98157773853484316821630077618, 6.10298626515778721430123935003, 6.28498501606813475465846017659
