L(s) = 1 | + 4·4-s + 4·9-s − 6·16-s + 16·36-s + 20·49-s − 60·64-s − 64·79-s − 6·81-s + 80·109-s + 56·121-s + 127-s + 131-s + 137-s + 139-s − 24·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + 40·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 80·196-s + ⋯ |
L(s) = 1 | + 2·4-s + 4/3·9-s − 3/2·16-s + 8/3·36-s + 20/7·49-s − 7.5·64-s − 7.20·79-s − 2/3·81-s + 7.66·109-s + 5.09·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 2·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 3.07·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 40/7·196-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.769542095\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.769542095\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 5 | \( 1 \) |
| 7 | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
good | 2 | \( ( 1 - T^{2} + p^{2} T^{4} )^{4} \) |
| 11 | \( ( 1 - 6 T + p T^{2} )^{4}( 1 + 6 T + p T^{2} )^{4} \) |
| 13 | \( ( 1 - 6 T + p T^{2} )^{4}( 1 + 6 T + p T^{2} )^{4} \) |
| 17 | \( ( 1 + 26 T^{2} + p^{2} T^{4} )^{4} \) |
| 19 | \( ( 1 - p T^{2} )^{8} \) |
| 23 | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{4} \) |
| 29 | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 + 34 T^{2} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 + p T^{2} )^{8} \) |
| 41 | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{4} \) |
| 43 | \( ( 1 + 62 T^{2} + p^{2} T^{4} )^{4} \) |
| 47 | \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{4} \) |
| 53 | \( ( 1 - p T^{2} )^{8} \) |
| 59 | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{4} \) |
| 61 | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 + 110 T^{2} + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 - 134 T^{2} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 + 8 T + p T^{2} )^{8} \) |
| 83 | \( ( 1 + 158 T^{2} + p^{2} T^{4} )^{4} \) |
| 89 | \( ( 1 + 70 T^{2} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 - 18 T + p T^{2} )^{4}( 1 + 18 T + p T^{2} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.62415214492046710740994592826, −4.56973722707512850517754720633, −4.54678262031045075492538122777, −4.34921749087422178565706792364, −4.32744118863675165866818591048, −4.10966826692915415762069196150, −4.08504269431150844708936455171, −3.72580293619557852195796827071, −3.58655451509827499408012405434, −3.25866290060073976563560065058, −3.25011780785064129175413650839, −3.13270138117997919483161469832, −3.06525395615780186104902710824, −2.62678131289774641261413808408, −2.61118554463342949171561613913, −2.42075279181311049976316458498, −2.37111464814532109163931501449, −2.07295131291423442897953309991, −1.88734868245612645954740273831, −1.67990065003813882294903981707, −1.61073099747690080640149575645, −1.60843867697175990493836109898, −0.956866069099139016628517934999, −0.73178242637463479010062257540, −0.36422796138032839689532786261,
0.36422796138032839689532786261, 0.73178242637463479010062257540, 0.956866069099139016628517934999, 1.60843867697175990493836109898, 1.61073099747690080640149575645, 1.67990065003813882294903981707, 1.88734868245612645954740273831, 2.07295131291423442897953309991, 2.37111464814532109163931501449, 2.42075279181311049976316458498, 2.61118554463342949171561613913, 2.62678131289774641261413808408, 3.06525395615780186104902710824, 3.13270138117997919483161469832, 3.25011780785064129175413650839, 3.25866290060073976563560065058, 3.58655451509827499408012405434, 3.72580293619557852195796827071, 4.08504269431150844708936455171, 4.10966826692915415762069196150, 4.32744118863675165866818591048, 4.34921749087422178565706792364, 4.54678262031045075492538122777, 4.56973722707512850517754720633, 4.62415214492046710740994592826
Plot not available for L-functions of degree greater than 10.