| L(s) = 1 | + 2·3-s − 2·7-s + 3·9-s − 4·11-s − 4·17-s + 4·19-s − 4·21-s − 8·23-s + 4·27-s − 8·29-s + 8·31-s − 8·33-s + 4·37-s − 16·41-s − 4·43-s − 4·47-s + 3·49-s − 8·51-s − 4·53-s + 8·57-s − 16·61-s − 6·63-s − 12·67-s − 16·69-s + 16·71-s − 24·73-s + 8·77-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.755·7-s + 9-s − 1.20·11-s − 0.970·17-s + 0.917·19-s − 0.872·21-s − 1.66·23-s + 0.769·27-s − 1.48·29-s + 1.43·31-s − 1.39·33-s + 0.657·37-s − 2.49·41-s − 0.609·43-s − 0.583·47-s + 3/7·49-s − 1.12·51-s − 0.549·53-s + 1.05·57-s − 2.04·61-s − 0.755·63-s − 1.46·67-s − 1.92·69-s + 1.89·71-s − 2.80·73-s + 0.911·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17640000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17640000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 11 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_4$ | \( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 8 T + 58 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 4 T + 58 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_4$ | \( 1 + 16 T + 126 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_4$ | \( 1 + 16 T + 166 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 12 T + 150 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 16 T + 186 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 24 T + 270 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 78 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 8 T + 14 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 16 T + 238 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
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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
−8.142602719090574167477286734427, −7.967427378184669324759081014128, −7.47780492166423502789757594684, −7.37174979309425825710423080861, −6.67713352177651971797603878102, −6.59326432327676294972712757421, −5.92173415471408485813214942890, −5.82516843263888697581511215778, −5.02322496286303741528382040670, −4.95993649845080315440281967670, −4.27534002761960482576162046688, −4.03199310661157672004602699429, −3.46009477911784801728057779914, −3.13078630377729798648968695269, −2.68141919270351332793159984935, −2.45405270193974583058545922645, −1.59146660648659462035110698696, −1.52643265782343537054209848181, 0, 0,
1.52643265782343537054209848181, 1.59146660648659462035110698696, 2.45405270193974583058545922645, 2.68141919270351332793159984935, 3.13078630377729798648968695269, 3.46009477911784801728057779914, 4.03199310661157672004602699429, 4.27534002761960482576162046688, 4.95993649845080315440281967670, 5.02322496286303741528382040670, 5.82516843263888697581511215778, 5.92173415471408485813214942890, 6.59326432327676294972712757421, 6.67713352177651971797603878102, 7.37174979309425825710423080861, 7.47780492166423502789757594684, 7.967427378184669324759081014128, 8.142602719090574167477286734427
