| L(s) = 1 | + 8·5-s − 12·13-s + 20·17-s + 38·25-s − 24·29-s − 12·37-s − 32·41-s − 20·53-s − 96·65-s − 20·73-s − 81-s + 160·85-s + 12·97-s − 16·109-s + 28·113-s + 20·121-s + 136·125-s + 127-s + 131-s + 137-s + 139-s − 192·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 3.57·5-s − 3.32·13-s + 4.85·17-s + 38/5·25-s − 4.45·29-s − 1.97·37-s − 4.99·41-s − 2.74·53-s − 11.9·65-s − 2.34·73-s − 1/9·81-s + 17.3·85-s + 1.21·97-s − 1.53·109-s + 2.63·113-s + 1.81·121-s + 12.1·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 15.9·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2500037663\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2500037663\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( 1 + T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| good | 7 | $C_2^3$ | \( 1 - 94 T^{4} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2}( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 23 | $C_2^3$ | \( 1 - 158 T^{4} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 43 | $C_2^3$ | \( 1 + 1202 T^{4} + p^{4} T^{8} \) |
| 47 | $C_2^3$ | \( 1 - 1918 T^{4} + p^{4} T^{8} \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2}( 1 + 14 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2}( 1 + 12 T + p T^{2} )^{2} \) |
| 67 | $C_2^3$ | \( 1 + 4946 T^{4} + p^{4} T^{8} \) |
| 71 | $C_2^2$ | \( ( 1 - 110 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 16 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 83 | $C_2^3$ | \( 1 - 13294 T^{4} + p^{4} T^{8} \) |
| 89 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−6.61568626765287622981600276759, −6.12216550651505874121068326797, −6.07278344419823668271149195784, −5.73463678624474268412850840589, −5.63946340764097627956386226692, −5.54239247078673681291059626217, −5.30047454010301453138406881305, −5.17215632767617599661915582820, −4.97271564187281920565508706509, −4.92146738121244052351036185268, −4.67515599524343816682096514028, −4.22956014479475581733902520196, −3.64171354712622111871275206164, −3.51518665897926578158381286700, −3.27647525801721334203075121597, −3.19351118537316870345001659675, −2.96028370757907681607453039673, −2.70681415432186837393293066592, −2.07575371651784362521431630525, −2.01937466697292819524265259815, −1.75926320571486989965816483907, −1.75125544801927630172107883135, −1.39330709911579664665272738313, −1.03228913426012835381701440636, −0.06610330703651128282527983553,
0.06610330703651128282527983553, 1.03228913426012835381701440636, 1.39330709911579664665272738313, 1.75125544801927630172107883135, 1.75926320571486989965816483907, 2.01937466697292819524265259815, 2.07575371651784362521431630525, 2.70681415432186837393293066592, 2.96028370757907681607453039673, 3.19351118537316870345001659675, 3.27647525801721334203075121597, 3.51518665897926578158381286700, 3.64171354712622111871275206164, 4.22956014479475581733902520196, 4.67515599524343816682096514028, 4.92146738121244052351036185268, 4.97271564187281920565508706509, 5.17215632767617599661915582820, 5.30047454010301453138406881305, 5.54239247078673681291059626217, 5.63946340764097627956386226692, 5.73463678624474268412850840589, 6.07278344419823668271149195784, 6.12216550651505874121068326797, 6.61568626765287622981600276759
