| L(s) = 1 | + 12·4-s − 20·16-s − 364·19-s − 588·31-s + 922·49-s + 1.70e3·61-s − 1.44e3·64-s − 4.36e3·76-s + 3.50e3·79-s + 3.24e3·109-s + 2.67e3·121-s − 7.05e3·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 6.33e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | + 3/2·4-s − 0.312·16-s − 4.39·19-s − 3.40·31-s + 2.68·49-s + 3.58·61-s − 2.81·64-s − 6.59·76-s + 4.99·79-s + 2.85·109-s + 2.01·121-s − 5.11·124-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 2.88·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.244560882\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.244560882\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( ( 1 - 3 p T^{2} + p^{6} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 461 T^{2} + p^{6} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 1338 T^{2} + p^{6} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 3169 T^{2} + p^{6} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 1986 T^{2} + p^{6} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 91 T + p^{3} T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 - 11374 T^{2} + p^{6} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 44778 T^{2} + p^{6} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 147 T + p^{3} T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 + 26 p^{2} T^{2} + p^{6} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 58158 T^{2} + p^{6} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 46789 T^{2} + p^{6} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 176286 T^{2} + p^{6} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 289914 T^{2} + p^{6} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 373242 T^{2} + p^{6} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 7 p T + p^{3} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 - 601301 T^{2} + p^{6} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 711822 T^{2} + p^{6} T^{4} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 - 773134 T^{2} + p^{6} T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 876 T + p^{3} T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 - 861334 T^{2} + p^{6} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 - 648121 T^{2} + p^{6} T^{4} )^{2} \) |
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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
−8.443434834850680752285119116856, −8.349394066343640826205752344731, −7.960969285221124169398637566560, −7.46135928998467572538510548669, −7.29784914026726195275685132207, −7.09600231809642229642619020899, −6.94346074939275645248964598673, −6.51156061203672841131135801078, −6.29775069127369547833828267176, −6.21144216567498654818943403921, −5.86598827803931735026584334301, −5.34509012550201650055277875612, −5.28892256675109539182266452110, −4.69474076164117674731632221597, −4.45860339148643725857178659170, −4.04624084963403353181420888668, −3.72526638631875332885558906131, −3.67673531946017526262660277463, −2.97943474939384084206824540315, −2.29879582618092112555991173880, −2.15565980484628573858913676225, −2.10944216568688078139541795858, −1.87105558585267892031174603272, −0.77728245128194881005115747777, −0.30324830918853772506236517044,
0.30324830918853772506236517044, 0.77728245128194881005115747777, 1.87105558585267892031174603272, 2.10944216568688078139541795858, 2.15565980484628573858913676225, 2.29879582618092112555991173880, 2.97943474939384084206824540315, 3.67673531946017526262660277463, 3.72526638631875332885558906131, 4.04624084963403353181420888668, 4.45860339148643725857178659170, 4.69474076164117674731632221597, 5.28892256675109539182266452110, 5.34509012550201650055277875612, 5.86598827803931735026584334301, 6.21144216567498654818943403921, 6.29775069127369547833828267176, 6.51156061203672841131135801078, 6.94346074939275645248964598673, 7.09600231809642229642619020899, 7.29784914026726195275685132207, 7.46135928998467572538510548669, 7.960969285221124169398637566560, 8.349394066343640826205752344731, 8.443434834850680752285119116856
