| L(s) = 1 | − 2-s + 4-s + 2·5-s − 3·8-s + 9-s − 2·10-s + 13-s + 16-s + 4·17-s − 18-s + 4·19-s + 2·20-s − 11·23-s + 2·25-s − 26-s − 3·29-s + 32-s − 4·34-s + 36-s − 3·37-s − 4·38-s − 6·40-s + 41-s + 6·43-s + 2·45-s + 11·46-s − 11·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.894·5-s − 1.06·8-s + 1/3·9-s − 0.632·10-s + 0.277·13-s + 1/4·16-s + 0.970·17-s − 0.235·18-s + 0.917·19-s + 0.447·20-s − 2.29·23-s + 2/5·25-s − 0.196·26-s − 0.557·29-s + 0.176·32-s − 0.685·34-s + 1/6·36-s − 0.493·37-s − 0.648·38-s − 0.948·40-s + 0.156·41-s + 0.914·43-s + 0.298·45-s + 1.62·46-s − 1.60·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5769 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5769 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7134288970\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7134288970\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 641 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 30 T + p T^{2} ) \) |
| good | 2 | $D_{4}$ | \( 1 + T + p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - T + 4 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 - 4 T + 14 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $D_{4}$ | \( 1 + 3 T + 28 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 3 T - 12 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - T + 24 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 47 | $D_{4}$ | \( 1 + 11 T + 78 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + T + 80 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 2 T + 86 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 12 T + 110 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 9 T + 148 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 9 T + 30 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 15 T + 178 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 4 T - 90 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 22 T + 274 T^{2} + 22 p T^{3} + p^{2} T^{4} \) |
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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
−17.7079767348, −16.7042948631, −16.3623680141, −15.9545810748, −15.4485351993, −14.6612282230, −14.3754782656, −13.6227734251, −13.3643718097, −12.4088589162, −12.0667906389, −11.5714515843, −10.7946393347, −10.1508304888, −9.70675962848, −9.37040765949, −8.59179579957, −7.94243997106, −7.33418112306, −6.43954778569, −5.92605913354, −5.32719039608, −4.04037278153, −2.97757118086, −1.77979110335,
1.77979110335, 2.97757118086, 4.04037278153, 5.32719039608, 5.92605913354, 6.43954778569, 7.33418112306, 7.94243997106, 8.59179579957, 9.37040765949, 9.70675962848, 10.1508304888, 10.7946393347, 11.5714515843, 12.0667906389, 12.4088589162, 13.3643718097, 13.6227734251, 14.3754782656, 14.6612282230, 15.4485351993, 15.9545810748, 16.3623680141, 16.7042948631, 17.7079767348
