| L(s) = 1 | + 9-s + 4·11-s + 4·16-s + 2·19-s − 16·29-s − 18·31-s − 40·41-s − 11·49-s − 24·59-s − 20·61-s − 24·71-s − 2·79-s + 32·89-s + 4·99-s − 4·101-s + 18·109-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 4·144-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 1/3·9-s + 1.20·11-s + 16-s + 0.458·19-s − 2.97·29-s − 3.23·31-s − 6.24·41-s − 1.57·49-s − 3.12·59-s − 2.56·61-s − 2.84·71-s − 0.225·79-s + 3.39·89-s + 0.402·99-s − 0.398·101-s + 1.72·109-s + 2.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1/3·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5436426655\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5436426655\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 5 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 + 11 T^{2} + p^{2} T^{4} \) |
| good | 2 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} )( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} ) \) |
| 11 | $C_2^2$ | \( ( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 25 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + p T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 + 9 T + 50 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^3$ | \( 1 + 65 T^{2} + 2856 T^{4} + 65 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 61 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 + 58 T^{2} + 1155 T^{4} + 58 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2^3$ | \( 1 - 38 T^{2} - 1365 T^{4} - 38 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + 10 T + 39 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 13 T^{2} + p^{2} T^{4} )( 1 + 122 T^{2} + p^{2} T^{4} ) \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 73 | $C_2^3$ | \( 1 + 137 T^{2} + 13440 T^{4} + 137 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 + T - 78 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 16 T + 167 T^{2} - 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2$ | \( ( 1 - 158 T^{2} + p^{2} T^{4} )^{2} \) |
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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
−7.79807115386811015684072114197, −7.77388255619842079995102857716, −7.16275999646881455723853421380, −7.09240027085019776790022268918, −7.03947466794383415308743400689, −6.79794039109062883203071575537, −6.28311243398311902774125173170, −6.15719604375254253271905355397, −5.80557135835649445070497932450, −5.71428031356590643755939812484, −5.38359078974667623584383529387, −5.17042692469687018896059269841, −4.76666755198047896407683086719, −4.53004630914722447925099297378, −4.44155531981708388385157319805, −3.82748842566449692845204616951, −3.42744419116655105625502276703, −3.34748726893631612254052168413, −3.32663180840104667562111604302, −3.13903205817527314071090966856, −2.01453804959049159936891766503, −1.70628688422667500909959272293, −1.64436792264599915532733731249, −1.60675948784508669872112413402, −0.22031265744412161086133938852,
0.22031265744412161086133938852, 1.60675948784508669872112413402, 1.64436792264599915532733731249, 1.70628688422667500909959272293, 2.01453804959049159936891766503, 3.13903205817527314071090966856, 3.32663180840104667562111604302, 3.34748726893631612254052168413, 3.42744419116655105625502276703, 3.82748842566449692845204616951, 4.44155531981708388385157319805, 4.53004630914722447925099297378, 4.76666755198047896407683086719, 5.17042692469687018896059269841, 5.38359078974667623584383529387, 5.71428031356590643755939812484, 5.80557135835649445070497932450, 6.15719604375254253271905355397, 6.28311243398311902774125173170, 6.79794039109062883203071575537, 7.03947466794383415308743400689, 7.09240027085019776790022268918, 7.16275999646881455723853421380, 7.77388255619842079995102857716, 7.79807115386811015684072114197
