| L(s) = 1 | + 2·5-s + 6·9-s + 4·13-s + 2·25-s − 8·29-s − 14·37-s − 18·41-s + 12·45-s − 28·53-s + 20·61-s + 8·65-s + 10·73-s + 27·81-s − 26·89-s + 26·97-s + 26·109-s + 32·113-s + 24·117-s + 10·125-s + 127-s + 131-s + 137-s + 139-s − 16·145-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 2·9-s + 1.10·13-s + 2/5·25-s − 1.48·29-s − 2.30·37-s − 2.81·41-s + 1.78·45-s − 3.84·53-s + 2.56·61-s + 0.992·65-s + 1.17·73-s + 3·81-s − 2.75·89-s + 2.63·97-s + 2.49·109-s + 3.01·113-s + 2.21·117-s + 0.894·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 1.32·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43264 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43264 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.810375262\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.810375262\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 13 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.72752473518288530742086238976, −12.43750271824789982129959033511, −11.44310348090023723243214984278, −11.29944957838954036334135690083, −10.39068003279472423351129738695, −10.32135201524751529429112683338, −9.637415616131866926976041594301, −9.471263247711531916231715033396, −8.520954952308135424933444597489, −8.388043132391436645426284328335, −7.29422603442057113574754261266, −7.15338587362535086062574500659, −6.42622687001759509914457494231, −6.04714819661523509914127030683, −4.97679577550062908376768076086, −4.94010679934541737022158512136, −3.64240142506372502278871588546, −3.55379710059607604052500440932, −1.88481271671562372321268480047, −1.57819236676149098736702579386,
1.57819236676149098736702579386, 1.88481271671562372321268480047, 3.55379710059607604052500440932, 3.64240142506372502278871588546, 4.94010679934541737022158512136, 4.97679577550062908376768076086, 6.04714819661523509914127030683, 6.42622687001759509914457494231, 7.15338587362535086062574500659, 7.29422603442057113574754261266, 8.388043132391436645426284328335, 8.520954952308135424933444597489, 9.471263247711531916231715033396, 9.637415616131866926976041594301, 10.32135201524751529429112683338, 10.39068003279472423351129738695, 11.29944957838954036334135690083, 11.44310348090023723243214984278, 12.43750271824789982129959033511, 12.72752473518288530742086238976
