| L(s) = 1 | + 4-s − 2·5-s − 8·13-s − 2·20-s + 2·25-s − 4·29-s − 2·37-s − 4·41-s − 8·52-s + 2·61-s − 64-s + 16·65-s − 2·73-s + 81-s − 4·89-s + 4·97-s + 2·100-s + 4·101-s − 2·109-s − 4·113-s − 4·116-s + 127-s + 131-s + 137-s + 139-s + 8·145-s − 2·148-s + ⋯ |
| L(s) = 1 | + 4-s − 2·5-s − 8·13-s − 2·20-s + 2·25-s − 4·29-s − 2·37-s − 4·41-s − 8·52-s + 2·61-s − 64-s + 16·65-s − 2·73-s + 81-s − 4·89-s + 4·97-s + 2·100-s + 4·101-s − 2·109-s − 4·113-s − 4·116-s + 127-s + 131-s + 137-s + 139-s + 8·145-s − 2·148-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{8} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{8} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.01217887186\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01217887186\) |
| \(L(1)\) |
|
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 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 7 | | \( 1 \) |
| 17 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| good | 3 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 5 | $C_2$$\times$$C_2^2$ | \( ( 1 + T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 11 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 13 | $C_1$ | \( ( 1 + T )^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 29 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \) |
| 31 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 37 | $C_2$$\times$$C_2^2$ | \( ( 1 + T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 + T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 61 | $C_2$$\times$$C_2^2$ | \( ( 1 - T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 73 | $C_2$$\times$$C_2^2$ | \( ( 1 + T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 79 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 89 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \) |
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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
−6.46371029834118770457696684905, −6.36398225657065211474226224729, −5.64849999007346632549395337950, −5.50658224206914421938532838479, −5.27939154624651060754916908468, −5.21753595963217796114579784231, −5.21035404823262886925983477800, −4.87986072673004156783776527319, −4.81782918993239323969554690927, −4.45418374612990984715044325718, −4.25499220940944368488979912829, −4.23187614255527727633377547713, −3.73938351189108354983259846442, −3.59763740073535095213674976767, −3.22644248481588589740173084569, −3.21686003581738107936935137237, −2.89198420898493790125070914590, −2.49266673648386482981755309533, −2.39310205260500398319360657231, −2.35716060353809213501587338268, −1.81961376715216428207290547883, −1.80712351809670074323291389806, −1.68018674049470842304377310802, −0.42880419490710514989288310188, −0.07646915520612395113461510800,
0.07646915520612395113461510800, 0.42880419490710514989288310188, 1.68018674049470842304377310802, 1.80712351809670074323291389806, 1.81961376715216428207290547883, 2.35716060353809213501587338268, 2.39310205260500398319360657231, 2.49266673648386482981755309533, 2.89198420898493790125070914590, 3.21686003581738107936935137237, 3.22644248481588589740173084569, 3.59763740073535095213674976767, 3.73938351189108354983259846442, 4.23187614255527727633377547713, 4.25499220940944368488979912829, 4.45418374612990984715044325718, 4.81782918993239323969554690927, 4.87986072673004156783776527319, 5.21035404823262886925983477800, 5.21753595963217796114579784231, 5.27939154624651060754916908468, 5.50658224206914421938532838479, 5.64849999007346632549395337950, 6.36398225657065211474226224729, 6.46371029834118770457696684905
