| L(s) = 1 | − 4·4-s − 4·7-s + 20·13-s + 12·16-s − 40·19-s + 46·25-s + 16·28-s − 76·31-s + 128·37-s + 92·43-s − 78·49-s − 80·52-s − 124·61-s − 32·64-s + 212·67-s − 208·73-s + 160·76-s − 28·79-s − 80·91-s − 28·97-s − 184·100-s − 148·103-s − 64·109-s − 48·112-s + 394·121-s + 304·124-s + 127-s + ⋯ |
| L(s) = 1 | − 4-s − 4/7·7-s + 1.53·13-s + 3/4·16-s − 2.10·19-s + 1.83·25-s + 4/7·28-s − 2.45·31-s + 3.45·37-s + 2.13·43-s − 1.59·49-s − 1.53·52-s − 2.03·61-s − 1/2·64-s + 3.16·67-s − 2.84·73-s + 2.10·76-s − 0.354·79-s − 0.879·91-s − 0.288·97-s − 1.83·100-s − 1.43·103-s − 0.587·109-s − 3/7·112-s + 3.25·121-s + 2.45·124-s + 0.00787·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.261439022\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.261439022\) |
| \(L(2)\) |
|
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 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( ( 1 - 23 T^{2} + p^{4} T^{4} )^{2} \) |
| 7 | $D_{4}$ | \( ( 1 + 2 T + 45 T^{2} + 2 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 394 T^{2} + 66147 T^{4} - 394 p^{4} T^{6} + p^{8} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 - 10 T + 147 T^{2} - 10 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 796 T^{2} + 294342 T^{4} - 796 p^{4} T^{6} + p^{8} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 20 T + 606 T^{2} + 20 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 2026 T^{2} + 1583907 T^{4} - 2026 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 3166 T^{2} + 3912675 T^{4} - 3166 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 38 T + 1797 T^{2} + 38 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 - 64 T + 3546 T^{2} - 64 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 2782 T^{2} + 4157187 T^{4} - 2782 p^{4} T^{6} + p^{8} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 46 T + 87 p T^{2} - 46 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 6586 T^{2} + 19745907 T^{4} - 6586 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 2236 T^{2} - 2409114 T^{4} - 2236 p^{4} T^{6} + p^{8} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 5194 T^{2} + 14880867 T^{4} - 5194 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 62 T + 6459 T^{2} + 62 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 - 106 T + 11301 T^{2} - 106 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 12460 T^{2} + 77194662 T^{4} - 12460 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 104 T + 11418 T^{2} + 104 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 + 14 T + 11181 T^{2} + 14 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 2842 T^{2} + 75503283 T^{4} - 2842 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 8860 T^{2} + 51019782 T^{4} - 8860 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 14 T + 8283 T^{2} + 14 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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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
−9.050836290853121295207364771755, −9.047720710102833870710738179347, −8.591780607957441458827971822548, −8.579326368995866687649677907975, −8.266578175984212976841839558238, −7.70588492282932355660362256315, −7.56024006225782581850414484474, −7.48410072990947499629169412342, −6.74848135309114683620825560554, −6.66525936196583481203423502646, −6.17530538694545667156514045317, −6.02265701632888676318450386899, −5.99430522267691183241930355741, −5.30326985027717009797753548261, −5.12828783582401275524614668286, −4.63858833647485661045741079928, −4.21335890700690418559036555164, −4.11882920190032061259256842791, −3.82741060615451024193347292959, −3.30351861002017920098741791142, −2.80290026511601776260126141880, −2.55596884653597274144918604449, −1.74040861290056030684881170202, −1.17591545205960433921518015879, −0.43740435151429851816795116332,
0.43740435151429851816795116332, 1.17591545205960433921518015879, 1.74040861290056030684881170202, 2.55596884653597274144918604449, 2.80290026511601776260126141880, 3.30351861002017920098741791142, 3.82741060615451024193347292959, 4.11882920190032061259256842791, 4.21335890700690418559036555164, 4.63858833647485661045741079928, 5.12828783582401275524614668286, 5.30326985027717009797753548261, 5.99430522267691183241930355741, 6.02265701632888676318450386899, 6.17530538694545667156514045317, 6.66525936196583481203423502646, 6.74848135309114683620825560554, 7.48410072990947499629169412342, 7.56024006225782581850414484474, 7.70588492282932355660362256315, 8.266578175984212976841839558238, 8.579326368995866687649677907975, 8.591780607957441458827971822548, 9.047720710102833870710738179347, 9.050836290853121295207364771755
