| L(s) = 1 | + 4-s − 9-s + 2·13-s + 2·17-s + 2·25-s − 2·29-s − 36-s + 4·37-s − 2·41-s + 2·52-s − 2·61-s − 64-s + 2·68-s + 2·73-s + 81-s + 2·89-s + 2·97-s + 2·100-s − 2·109-s + 8·113-s − 2·116-s − 2·117-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 4-s − 9-s + 2·13-s + 2·17-s + 2·25-s − 2·29-s − 36-s + 4·37-s − 2·41-s + 2·52-s − 2·61-s − 64-s + 2·68-s + 2·73-s + 81-s + 2·89-s + 2·97-s + 2·100-s − 2·109-s + 8·113-s − 2·116-s − 2·117-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(3.224230539\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.224230539\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 19 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| good | 3 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 5 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 23 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 29 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
| 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_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 53 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 59 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 89 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
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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
−6.02189186281492031888359515432, −5.91989624513621390697092791723, −5.87750198315235327173659525833, −5.73484887567308191455206168245, −5.73413210212115335758353735865, −5.25216104936584938997412715315, −4.91036020281776479070697520718, −4.84683908691249766018604418907, −4.68627806071216564871269862224, −4.47272940884091267071342294289, −4.27026675870591517719560576562, −3.68553179156600485277215250780, −3.62466116095247499080432083322, −3.52745981939976592603825745047, −3.40100107124991575497294587643, −3.00660564358501641520362332580, −2.99915762846065234162644832217, −2.72956116339424830848870014805, −2.19010325190831402524207284848, −2.17293044368975333148887017350, −1.96731395285619897931016149609, −1.56615446480715772972578179219, −1.16019925302006036032582390610, −0.945419366990346898714907974535, −0.77473787396442241495395187797,
0.77473787396442241495395187797, 0.945419366990346898714907974535, 1.16019925302006036032582390610, 1.56615446480715772972578179219, 1.96731395285619897931016149609, 2.17293044368975333148887017350, 2.19010325190831402524207284848, 2.72956116339424830848870014805, 2.99915762846065234162644832217, 3.00660564358501641520362332580, 3.40100107124991575497294587643, 3.52745981939976592603825745047, 3.62466116095247499080432083322, 3.68553179156600485277215250780, 4.27026675870591517719560576562, 4.47272940884091267071342294289, 4.68627806071216564871269862224, 4.84683908691249766018604418907, 4.91036020281776479070697520718, 5.25216104936584938997412715315, 5.73413210212115335758353735865, 5.73484887567308191455206168245, 5.87750198315235327173659525833, 5.91989624513621390697092791723, 6.02189186281492031888359515432
