| L(s) = 1 | − 100·25-s + 188·49-s − 184·73-s + 8·97-s + 484·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 292·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + ⋯ |
| L(s) = 1 | − 4·25-s + 3.83·49-s − 2.52·73-s + 8/97·97-s + 4·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 1.72·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + 0.00473·211-s + 0.00448·223-s + 0.00440·227-s + 0.00436·229-s + 0.00429·233-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.441944007\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.441944007\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 7 | $C_2^2$ | \( ( 1 - 94 T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 13 | $C_2^2$ | \( ( 1 + 146 T^{2} + p^{4} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 19 | $C_2$ | \( ( 1 - 26 T + p^{2} T^{2} )^{2}( 1 + 26 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 29 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 + 194 T^{2} + p^{4} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - 2062 T^{2} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 43 | $C_2$ | \( ( 1 - 22 T + p^{2} T^{2} )^{2}( 1 + 22 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 53 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 61 | $C_2^2$ | \( ( 1 - 1966 T^{2} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 122 T + p^{2} T^{2} )^{2}( 1 + 122 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 73 | $C_2$ | \( ( 1 + 46 T + p^{2} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 7682 T^{2} + p^{4} T^{4} )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{4}( 1 + p T )^{4} \) |
| 89 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p^{2} T^{2} )^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.05007662987097755302700461408, −5.92868798608084289000154036913, −5.82287087430057570229943984018, −5.68998749069041600488985731644, −5.56549297165530925564399277277, −5.21150718482105195636687613336, −4.89221071113872100082874327917, −4.59762212803831044866636257056, −4.54352273170685158708011796442, −4.37128960434190520622242616431, −3.87864323505332819700706137118, −3.81463205791533725876342778387, −3.73360162475811695293308427208, −3.56376982712620355006861496839, −3.13138767326344879571441330650, −2.75340168926415540908018792907, −2.64270932463618340768017923252, −2.33901227208363476252769218598, −2.09789687933273866820372978479, −1.80454615586364721233489680523, −1.69681759386098333015303043424, −1.13699750212752712994229667093, −0.993176949362384907866768054133, −0.38082544349490654396360220884, −0.28621413222160152279031065412,
0.28621413222160152279031065412, 0.38082544349490654396360220884, 0.993176949362384907866768054133, 1.13699750212752712994229667093, 1.69681759386098333015303043424, 1.80454615586364721233489680523, 2.09789687933273866820372978479, 2.33901227208363476252769218598, 2.64270932463618340768017923252, 2.75340168926415540908018792907, 3.13138767326344879571441330650, 3.56376982712620355006861496839, 3.73360162475811695293308427208, 3.81463205791533725876342778387, 3.87864323505332819700706137118, 4.37128960434190520622242616431, 4.54352273170685158708011796442, 4.59762212803831044866636257056, 4.89221071113872100082874327917, 5.21150718482105195636687613336, 5.56549297165530925564399277277, 5.68998749069041600488985731644, 5.82287087430057570229943984018, 5.92868798608084289000154036913, 6.05007662987097755302700461408
