| L(s) = 1 | + 2·5-s − 9-s − 2·13-s − 2·17-s + 3·25-s + 2·37-s − 2·41-s − 2·45-s − 49-s − 2·53-s − 4·65-s − 2·73-s + 81-s − 4·85-s + 2·113-s + 2·117-s − 121-s + 6·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 2·153-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | + 2·5-s − 9-s − 2·13-s − 2·17-s + 3·25-s + 2·37-s − 2·41-s − 2·45-s − 49-s − 2·53-s − 4·65-s − 2·73-s + 81-s − 4·85-s + 2·113-s + 2·117-s − 121-s + 6·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 2·153-s + 157-s + 163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9821551531\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9821551531\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 31 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| good | 3 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 7 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 11 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 19 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 29 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 43 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 53 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 59 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 61 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 67 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 71 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 79 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 83 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
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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
−6.58277062721921466378865427556, −6.44890806638460420342135796666, −6.26922485789195572822733717201, −6.21066078101829189244331727224, −5.97440261911471317640928119174, −5.51451975976903869881156777510, −5.46125830841240864794367092631, −5.41311767434470613029956186916, −4.94375045952144991524498094054, −4.80100663378862606117118451118, −4.64277121032283770493315847182, −4.59502158495483365140477653730, −4.20099839177430129348746323847, −4.04920243340125605272452847751, −3.46588726751270915960862410420, −3.22685080572336038712856775444, −3.06061208596065407044108493998, −2.76727707435279128502493559731, −2.70762004757654679328877032014, −2.23123348947667574866130524240, −2.09132947034359915894078793871, −1.95544786336614072643194391488, −1.56709422524660298961366664399, −1.22162048872448090732152873889, −0.46337993976328213097538312990,
0.46337993976328213097538312990, 1.22162048872448090732152873889, 1.56709422524660298961366664399, 1.95544786336614072643194391488, 2.09132947034359915894078793871, 2.23123348947667574866130524240, 2.70762004757654679328877032014, 2.76727707435279128502493559731, 3.06061208596065407044108493998, 3.22685080572336038712856775444, 3.46588726751270915960862410420, 4.04920243340125605272452847751, 4.20099839177430129348746323847, 4.59502158495483365140477653730, 4.64277121032283770493315847182, 4.80100663378862606117118451118, 4.94375045952144991524498094054, 5.41311767434470613029956186916, 5.46125830841240864794367092631, 5.51451975976903869881156777510, 5.97440261911471317640928119174, 6.21066078101829189244331727224, 6.26922485789195572822733717201, 6.44890806638460420342135796666, 6.58277062721921466378865427556
