| L(s) = 1 | − 3·9-s − 18·19-s + 36·31-s + 13·49-s − 30·61-s + 34·79-s − 34·109-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 46·169-s + 54·171-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | − 9-s − 4.12·19-s + 6.46·31-s + 13/7·49-s − 3.84·61-s + 3.82·79-s − 3.25·109-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 3.53·169-s + 4.12·171-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5016573850\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5016573850\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 5 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| good | 11 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 23 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2}( 1 - 7 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 73 T^{2} + p^{2} T^{4} )( 1 + 26 T^{2} + p^{2} T^{4} ) \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T + p T^{2} )^{2}( 1 + 14 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 109 T^{2} + p^{2} T^{4} )( 1 + 122 T^{2} + p^{2} T^{4} ) \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 73 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 97 T^{2} + p^{2} T^{4} )( 1 - 46 T^{2} + p^{2} T^{4} ) \) |
| 79 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2}( 1 - 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 89 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 169 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−6.52524163293491696802384785619, −6.41468880150551895409943495839, −6.11410658632003400538905021699, −5.97005681110183550646598519654, −5.64056368436443038121618241452, −5.55528232701351724497699340138, −5.12427363764397657723481588770, −4.93133261247043515302946876259, −4.65658897799173012480975950151, −4.39335674989657498942100533399, −4.36546782644587033812636487668, −4.35957753746558317742316602531, −3.94611131814498482222822657044, −3.71035757062377461355335553867, −3.26338426665225181151839098670, −3.03386024321708511224590664998, −2.82487513137547798235875120675, −2.60528663132931696460484937669, −2.39008828539541413001348419585, −2.20328068816299253490870850841, −1.92031155015977353708394047005, −1.42025949416906607433129148542, −0.975921734353406488605339778144, −0.818643975557155154483942556960, −0.13663626241023770969549641904,
0.13663626241023770969549641904, 0.818643975557155154483942556960, 0.975921734353406488605339778144, 1.42025949416906607433129148542, 1.92031155015977353708394047005, 2.20328068816299253490870850841, 2.39008828539541413001348419585, 2.60528663132931696460484937669, 2.82487513137547798235875120675, 3.03386024321708511224590664998, 3.26338426665225181151839098670, 3.71035757062377461355335553867, 3.94611131814498482222822657044, 4.35957753746558317742316602531, 4.36546782644587033812636487668, 4.39335674989657498942100533399, 4.65658897799173012480975950151, 4.93133261247043515302946876259, 5.12427363764397657723481588770, 5.55528232701351724497699340138, 5.64056368436443038121618241452, 5.97005681110183550646598519654, 6.11410658632003400538905021699, 6.41468880150551895409943495839, 6.52524163293491696802384785619
