| L(s) = 1 | + 2-s − 8-s − 9-s − 16-s − 18-s − 2·23-s − 2·25-s + 2·31-s + 2·41-s − 2·46-s − 2·47-s + 2·49-s − 2·50-s + 2·62-s + 64-s − 2·71-s + 72-s + 2·73-s + 2·82-s − 2·94-s + 2·98-s − 2·121-s + 127-s + 128-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | + 2-s − 8-s − 9-s − 16-s − 18-s − 2·23-s − 2·25-s + 2·31-s + 2·41-s − 2·46-s − 2·47-s + 2·49-s − 2·50-s + 2·62-s + 64-s − 2·71-s + 72-s + 2·73-s + 2·82-s − 2·94-s + 2·98-s − 2·121-s + 127-s + 128-s + 131-s + 137-s + 139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33856 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33856 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6093169568\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6093169568\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 23 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 11 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 19 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 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
−13.10995043838533892545172818669, −12.73450827178459867040573226024, −11.94239805290365940957762084181, −11.69243299869226911288025580321, −11.68149406203610723558258075500, −10.64299439422787121241737197914, −10.20868582024761209641104733360, −9.488495768896575088723318240994, −9.319536142440528921527770208754, −8.301570214589871528252209263656, −8.206175263689184412407679820488, −7.56162980886532190479026971444, −6.57311791696696270035058239893, −5.94437251995716875291301790728, −5.88206029672434117157282978520, −5.06483423795029329512520436940, −4.21028620489519597444489717303, −3.93292280377889332414245401166, −2.94719270292561927370400946891, −2.25198209300471212712706150231,
2.25198209300471212712706150231, 2.94719270292561927370400946891, 3.93292280377889332414245401166, 4.21028620489519597444489717303, 5.06483423795029329512520436940, 5.88206029672434117157282978520, 5.94437251995716875291301790728, 6.57311791696696270035058239893, 7.56162980886532190479026971444, 8.206175263689184412407679820488, 8.301570214589871528252209263656, 9.319536142440528921527770208754, 9.488495768896575088723318240994, 10.20868582024761209641104733360, 10.64299439422787121241737197914, 11.68149406203610723558258075500, 11.69243299869226911288025580321, 11.94239805290365940957762084181, 12.73450827178459867040573226024, 13.10995043838533892545172818669
