| L(s) = 1 | − 2·3-s + 2·5-s + 3·9-s − 8·11-s − 2·13-s − 4·15-s + 4·17-s + 3·25-s − 4·27-s − 4·29-s + 8·31-s + 16·33-s + 4·37-s + 4·39-s + 12·41-s − 8·43-s + 6·45-s − 8·47-s − 6·49-s − 8·51-s + 4·53-s − 16·55-s − 8·59-s − 4·61-s − 4·65-s − 16·71-s + 12·73-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.894·5-s + 9-s − 2.41·11-s − 0.554·13-s − 1.03·15-s + 0.970·17-s + 3/5·25-s − 0.769·27-s − 0.742·29-s + 1.43·31-s + 2.78·33-s + 0.657·37-s + 0.640·39-s + 1.87·41-s − 1.21·43-s + 0.894·45-s − 1.16·47-s − 6/7·49-s − 1.12·51-s + 0.549·53-s − 2.15·55-s − 1.04·59-s − 0.512·61-s − 0.496·65-s − 1.89·71-s + 1.40·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38937600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38937600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 8 T + 46 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_4$ | \( 1 - 12 T + 86 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 4 T - 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 - 12 T + 174 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 126 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 8 T + 54 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 4 T + 150 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 20 T + 222 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
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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
−7.71981093309745841320485582719, −7.69857981961503792669088667116, −7.01733963206345011554946556651, −6.88815625544501699846327504972, −6.19859125501433445460824016676, −6.16020434400621363874343855175, −5.56534966430287245836261513492, −5.49451779205494916079187629934, −5.06839996635424300269417879593, −4.85050535994677646438217569339, −4.37713589862849578523092936947, −4.02942235834725422079721166916, −3.09048257757754632206312473517, −3.06763354007824267055826391951, −2.46412081208854751346991094724, −2.20756887601342391484529583850, −1.36163877886909068404871055992, −1.14360649981184563108810772519, 0, 0,
1.14360649981184563108810772519, 1.36163877886909068404871055992, 2.20756887601342391484529583850, 2.46412081208854751346991094724, 3.06763354007824267055826391951, 3.09048257757754632206312473517, 4.02942235834725422079721166916, 4.37713589862849578523092936947, 4.85050535994677646438217569339, 5.06839996635424300269417879593, 5.49451779205494916079187629934, 5.56534966430287245836261513492, 6.16020434400621363874343855175, 6.19859125501433445460824016676, 6.88815625544501699846327504972, 7.01733963206345011554946556651, 7.69857981961503792669088667116, 7.71981093309745841320485582719
