| L(s) = 1 | + 6·3-s − 4·4-s + 27·9-s − 24·12-s + 52·13-s + 16·16-s + 42·17-s − 6·23-s − 25·25-s + 108·27-s − 300·29-s − 108·36-s + 312·39-s + 4·43-s + 96·48-s + 245·49-s + 252·51-s − 208·52-s + 894·53-s − 886·61-s − 64·64-s − 168·68-s − 36·69-s − 150·75-s − 1.57e3·79-s + 405·81-s − 1.80e3·87-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1/2·4-s + 9-s − 0.577·12-s + 1.10·13-s + 1/4·16-s + 0.599·17-s − 0.0543·23-s − 1/5·25-s + 0.769·27-s − 1.92·29-s − 1/2·36-s + 1.28·39-s + 0.0141·43-s + 0.288·48-s + 5/7·49-s + 0.691·51-s − 0.554·52-s + 2.31·53-s − 1.85·61-s − 1/8·64-s − 0.299·68-s − 0.0628·69-s − 0.230·75-s − 2.23·79-s + 5/9·81-s − 2.21·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 152100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 152100 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.756983888\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.756983888\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 3 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 5 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 13 | $C_2$ | \( 1 - 4 p T + p^{3} T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 5 p^{2} T^{2} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2437 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 21 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 12422 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 3 T + p^{3} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 150 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 15482 T^{2} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 72065 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 56617 T^{2} + p^{6} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 2 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 124130 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 447 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 369142 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 443 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 269750 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 704797 T^{2} + p^{6} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 698510 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 785 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 1026610 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 1332097 T^{2} + p^{6} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 639425 T^{2} + p^{6} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.31466142566558966379507297098, −10.52875201866823944434971718230, −9.940179218385933004076264527871, −9.902914513578465079582804943038, −8.997132660751560081998369491677, −8.873515156137270592779863134498, −8.592597290774731392726164131327, −7.76424350826318256011329187961, −7.61601341201508198380493962156, −7.08169924440893244479496822581, −6.37887576066394819279149629837, −5.60528102550131536711629541996, −5.54800051571303526719645120893, −4.39555004272049863095840289612, −4.16021601094398458810829103730, −3.44197932254096202426045867968, −3.11153290162452249504144572542, −2.13988961562106245979747111579, −1.53588155549046645660625891735, −0.63631574878108170554726116534,
0.63631574878108170554726116534, 1.53588155549046645660625891735, 2.13988961562106245979747111579, 3.11153290162452249504144572542, 3.44197932254096202426045867968, 4.16021601094398458810829103730, 4.39555004272049863095840289612, 5.54800051571303526719645120893, 5.60528102550131536711629541996, 6.37887576066394819279149629837, 7.08169924440893244479496822581, 7.61601341201508198380493962156, 7.76424350826318256011329187961, 8.592597290774731392726164131327, 8.873515156137270592779863134498, 8.997132660751560081998369491677, 9.902914513578465079582804943038, 9.940179218385933004076264527871, 10.52875201866823944434971718230, 11.31466142566558966379507297098
