L(s) = 1 | + 3-s + 19-s + 25-s − 27-s − 31-s − 37-s + 57-s + 3·67-s + 3·73-s + 75-s − 3·79-s − 81-s − 93-s − 103-s + 109-s − 111-s + 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + ⋯ |
L(s) = 1 | + 3-s + 19-s + 25-s − 27-s − 31-s − 37-s + 57-s + 3·67-s + 3·73-s + 75-s − 3·79-s − 81-s − 93-s − 103-s + 109-s − 111-s + 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.766636660\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.766636660\) |
\(L(1)\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | | \( 1 \) |
good | 5 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 11 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 - T + T^{2} ) \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 + T + T^{2} ) \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.386725719406232762259228325723, −8.829431542911691174532891929046, −8.632883593553366184180235509258, −8.285130916565853550874192986572, −7.943640985036925301273516970132, −7.44131565601255858007011060816, −7.10305047058641789067305016253, −6.85300142354819138364791964514, −6.30792394344717580039821369900, −5.79350615601559685751274934500, −5.29843358412518647895401140583, −5.16571731499131676393689109910, −4.57264677772177362081331077248, −3.82989702312932093486584324411, −3.72928945178457173015163671959, −3.11110430347519644808400840119, −2.81490086691775600320431139664, −2.15346985224634997342782038706, −1.74336759877131082053515560156, −0.876804002733165351353712806347,
0.876804002733165351353712806347, 1.74336759877131082053515560156, 2.15346985224634997342782038706, 2.81490086691775600320431139664, 3.11110430347519644808400840119, 3.72928945178457173015163671959, 3.82989702312932093486584324411, 4.57264677772177362081331077248, 5.16571731499131676393689109910, 5.29843358412518647895401140583, 5.79350615601559685751274934500, 6.30792394344717580039821369900, 6.85300142354819138364791964514, 7.10305047058641789067305016253, 7.44131565601255858007011060816, 7.943640985036925301273516970132, 8.285130916565853550874192986572, 8.632883593553366184180235509258, 8.829431542911691174532891929046, 9.386725719406232762259228325723