| L(s) = 1 | − 5-s + 2·9-s + 11-s − 13-s + 19-s + 23-s + 31-s − 2·45-s + 2·49-s − 55-s + 65-s + 67-s − 73-s − 2·79-s + 3·81-s − 4·83-s − 89-s − 95-s − 97-s + 2·99-s − 101-s − 115-s − 2·117-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 5-s + 2·9-s + 11-s − 13-s + 19-s + 23-s + 31-s − 2·45-s + 2·49-s − 55-s + 65-s + 67-s − 73-s − 2·79-s + 3·81-s − 4·83-s − 89-s − 95-s − 97-s + 2·99-s − 101-s − 115-s − 2·117-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1597696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1597696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.133916854\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.133916854\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 79 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 5 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 11 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 13 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 19 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 23 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 67 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 83 | $C_1$ | \( ( 1 + T )^{4} \) |
| 89 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 97 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
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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
−9.907135892367631838250896954216, −9.829801807003953157478102487683, −9.325465599260351693255651923396, −8.967700477899938268555759518315, −8.294433615856057765424289976892, −8.154355668905124952404274844364, −7.30659079279449025668433750082, −7.25544407963965417200099046680, −7.05387530454843688936240317915, −6.65483324343715194501014118334, −5.80660239597333349970510981606, −5.53287242584438726055711694532, −4.64923839918670576285285811113, −4.60149991100736925799271846309, −4.01921444231014817335883723546, −3.78371449567357649697179012086, −2.99073913439453756868555005967, −2.51719698111865183022931018397, −1.48750417157765528420555652390, −1.11475419861668747198982774317,
1.11475419861668747198982774317, 1.48750417157765528420555652390, 2.51719698111865183022931018397, 2.99073913439453756868555005967, 3.78371449567357649697179012086, 4.01921444231014817335883723546, 4.60149991100736925799271846309, 4.64923839918670576285285811113, 5.53287242584438726055711694532, 5.80660239597333349970510981606, 6.65483324343715194501014118334, 7.05387530454843688936240317915, 7.25544407963965417200099046680, 7.30659079279449025668433750082, 8.154355668905124952404274844364, 8.294433615856057765424289976892, 8.967700477899938268555759518315, 9.325465599260351693255651923396, 9.829801807003953157478102487683, 9.907135892367631838250896954216
