| L(s) = 1 | + 4-s + 9-s + 2·11-s − 8·19-s + 36-s − 4·41-s + 2·44-s + 2·49-s − 2·59-s − 64-s − 8·76-s + 4·89-s + 2·99-s + 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s − 4·164-s + 167-s + 2·169-s − 8·171-s + 173-s + ⋯ |
| L(s) = 1 | + 4-s + 9-s + 2·11-s − 8·19-s + 36-s − 4·41-s + 2·44-s + 2·49-s − 2·59-s − 64-s − 8·76-s + 4·89-s + 2·99-s + 3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s − 4·164-s + 167-s + 2·169-s − 8·171-s + 173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9506311319\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9506311319\) |
| \(L(1)\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 3 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 19 | $C_1$ | \( ( 1 + T )^{8} \) |
| 23 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 41 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
| 43 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 47 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 59 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 73 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 83 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
| 97 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
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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
−6.72438027726173487667877908285, −6.56875302196552206070886735752, −6.35932534241057453707300763664, −6.30625675643013660697211699987, −6.16901194325622101442255250526, −6.06203510735435564077399046923, −5.71416103453631749697386136369, −5.12881101680909443395623677804, −5.01416789179383666717981765888, −4.87113747219715815344168604260, −4.46176799045250495728152669080, −4.28594526994680840127388930445, −4.17447292961991938419969427431, −4.10187096955129550011389501728, −3.80289844165556543901472994640, −3.56601698822310785228126100365, −3.32332478407927401159022863107, −2.76594477286899714820133867788, −2.56859065300355188838635233570, −2.25859564692299210050947729146, −1.97058776441899405761209405845, −1.77935328095653857602908539241, −1.76793121082152067002823293072, −1.42860540125976050099974465698, −0.45395563423841967327361832375,
0.45395563423841967327361832375, 1.42860540125976050099974465698, 1.76793121082152067002823293072, 1.77935328095653857602908539241, 1.97058776441899405761209405845, 2.25859564692299210050947729146, 2.56859065300355188838635233570, 2.76594477286899714820133867788, 3.32332478407927401159022863107, 3.56601698822310785228126100365, 3.80289844165556543901472994640, 4.10187096955129550011389501728, 4.17447292961991938419969427431, 4.28594526994680840127388930445, 4.46176799045250495728152669080, 4.87113747219715815344168604260, 5.01416789179383666717981765888, 5.12881101680909443395623677804, 5.71416103453631749697386136369, 6.06203510735435564077399046923, 6.16901194325622101442255250526, 6.30625675643013660697211699987, 6.35932534241057453707300763664, 6.56875302196552206070886735752, 6.72438027726173487667877908285
