| L(s) = 1 | − 2·3-s − 4-s − 3·7-s + 2·9-s + 2·11-s + 2·12-s + 2·13-s + 16-s − 2·17-s + 6·21-s + 3·23-s − 4·25-s − 6·27-s + 3·28-s + 8·29-s − 2·31-s − 4·33-s − 2·36-s − 6·37-s − 4·39-s + 2·41-s + 2·43-s − 2·44-s + 4·47-s − 2·48-s + 4·51-s − 2·52-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1/2·4-s − 1.13·7-s + 2/3·9-s + 0.603·11-s + 0.577·12-s + 0.554·13-s + 1/4·16-s − 0.485·17-s + 1.30·21-s + 0.625·23-s − 4/5·25-s − 1.15·27-s + 0.566·28-s + 1.48·29-s − 0.359·31-s − 0.696·33-s − 1/3·36-s − 0.986·37-s − 0.640·39-s + 0.312·41-s + 0.304·43-s − 0.301·44-s + 0.583·47-s − 0.288·48-s + 0.560·51-s − 0.277·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3087021439\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3087021439\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 4 T + p T^{2} ) \) |
| good | 3 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $D_{4}$ | \( 1 + 2 T - 6 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 12 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + p T^{2} ) \) |
| 31 | $D_{4}$ | \( 1 + 2 T - 26 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 6 T + 22 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $D_{4}$ | \( 1 - 2 T + 18 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 2 T + 46 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $D_{4}$ | \( 1 - 10 T + 78 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 4 T + 44 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 90 T^{2} + p^{2} T^{4} \) |
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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
−19.4237963090, −19.1492733656, −18.4313169827, −17.7791037642, −17.3057642744, −16.8749677391, −16.1071861552, −15.7309621332, −15.0375848666, −13.9804591873, −13.5544476283, −12.7876055630, −12.2653252030, −11.5603279670, −10.8544583040, −10.1370351403, −9.37368605065, −8.74377833705, −7.48523119168, −6.45298028941, −6.00213247982, −4.82710342129, −3.62441303496,
3.62441303496, 4.82710342129, 6.00213247982, 6.45298028941, 7.48523119168, 8.74377833705, 9.37368605065, 10.1370351403, 10.8544583040, 11.5603279670, 12.2653252030, 12.7876055630, 13.5544476283, 13.9804591873, 15.0375848666, 15.7309621332, 16.1071861552, 16.8749677391, 17.3057642744, 17.7791037642, 18.4313169827, 19.1492733656, 19.4237963090
