| L(s) = 1 | − 2·4-s + 2·5-s + 9-s + 10·11-s + 3·16-s + 8·19-s − 4·20-s + 5·25-s + 8·29-s − 8·31-s − 2·36-s − 20·44-s + 2·45-s − 10·49-s + 20·55-s + 8·59-s − 8·61-s − 4·64-s − 16·76-s + 10·79-s + 6·80-s + 16·95-s + 10·99-s − 10·100-s + 68·101-s + 32·109-s − 16·116-s + ⋯ |
| L(s) = 1 | − 4-s + 0.894·5-s + 1/3·9-s + 3.01·11-s + 3/4·16-s + 1.83·19-s − 0.894·20-s + 25-s + 1.48·29-s − 1.43·31-s − 1/3·36-s − 3.01·44-s + 0.298·45-s − 1.42·49-s + 2.69·55-s + 1.04·59-s − 1.02·61-s − 1/2·64-s − 1.83·76-s + 1.12·79-s + 0.670·80-s + 1.64·95-s + 1.00·99-s − 100-s + 6.76·101-s + 3.06·109-s − 1.48·116-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.469653494\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.469653494\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 3 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| good | 7 | $C_2^3$ | \( 1 + 10 T^{2} + 51 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 - 5 T + 14 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - T^{2} + p^{2} T^{4} )( 1 + 23 T^{2} + p^{2} T^{4} ) \) |
| 17 | $C_2^3$ | \( 1 + 25 T^{2} + 336 T^{4} + 25 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 37 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{4} \) |
| 37 | $C_2^3$ | \( 1 + 49 T^{2} + 1032 T^{4} + 49 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^3$ | \( 1 + 85 T^{2} + 5376 T^{4} + 85 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 - 13 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^3$ | \( 1 + 42 T^{2} - 1045 T^{4} + 42 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 - 4 T - 43 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 67 | $C_2^3$ | \( 1 + 85 T^{2} + 2736 T^{4} + 85 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 16 T + 183 T^{2} - 16 p T^{3} + p^{2} T^{4} )( 1 + 16 T + 183 T^{2} + 16 p T^{3} + p^{2} T^{4} ) \) |
| 79 | $C_2^2$ | \( ( 1 - 5 T - 54 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 + 130 T^{2} + 10011 T^{4} + 130 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
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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
−7.22536084614438592387218736465, −7.07983885016726353725293255668, −6.53597534612919382666844838612, −6.43920596594223751253929865015, −6.43316159839270667801433492934, −6.17065328445446170290452624816, −5.94703428127032476620971602910, −5.47455130463277163614346925706, −5.44674407037360427981728807080, −5.02913024755298323214525317011, −4.97313790205768912551746053087, −4.59253725632468668622997870422, −4.42521739603077710208987768990, −4.08357345122381079003775041300, −4.03382767497262426722460762960, −3.39793531462044071065179779475, −3.32587196482208242300162710082, −3.24477575697928932429719072066, −3.02355045364489089186911257756, −2.12658526572551069913824526040, −1.98645003579625932602866018188, −1.82955640197541103442578396430, −1.14137926353891839772924293964, −1.02631947495674979241166859843, −0.70305337668896462908401572699,
0.70305337668896462908401572699, 1.02631947495674979241166859843, 1.14137926353891839772924293964, 1.82955640197541103442578396430, 1.98645003579625932602866018188, 2.12658526572551069913824526040, 3.02355045364489089186911257756, 3.24477575697928932429719072066, 3.32587196482208242300162710082, 3.39793531462044071065179779475, 4.03382767497262426722460762960, 4.08357345122381079003775041300, 4.42521739603077710208987768990, 4.59253725632468668622997870422, 4.97313790205768912551746053087, 5.02913024755298323214525317011, 5.44674407037360427981728807080, 5.47455130463277163614346925706, 5.94703428127032476620971602910, 6.17065328445446170290452624816, 6.43316159839270667801433492934, 6.43920596594223751253929865015, 6.53597534612919382666844838612, 7.07983885016726353725293255668, 7.22536084614438592387218736465
