| L(s) = 1 | − 3·5-s + 7-s − 3·9-s − 3·11-s + 13-s + 12·17-s − 8·19-s + 3·23-s + 5·25-s − 3·29-s − 5·31-s − 3·35-s + 4·37-s − 3·41-s + 43-s + 9·45-s + 9·47-s + 7·49-s − 12·53-s + 9·55-s + 3·59-s + 13·61-s − 3·63-s − 3·65-s + 7·67-s − 24·71-s − 20·73-s + ⋯ |
| L(s) = 1 | − 1.34·5-s + 0.377·7-s − 9-s − 0.904·11-s + 0.277·13-s + 2.91·17-s − 1.83·19-s + 0.625·23-s + 25-s − 0.557·29-s − 0.898·31-s − 0.507·35-s + 0.657·37-s − 0.468·41-s + 0.152·43-s + 1.34·45-s + 1.31·47-s + 49-s − 1.64·53-s + 1.21·55-s + 0.390·59-s + 1.66·61-s − 0.377·63-s − 0.372·65-s + 0.855·67-s − 2.84·71-s − 2.34·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4840633851\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4840633851\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 3 T + 4 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 3 T - 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 5 T - 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 9 T + 34 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 3 T - 50 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 7 T - 18 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 11 T + 42 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 9 T - 2 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 11 T + 24 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
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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
−16.53794698895738274335983513303, −16.52017147857990237778726022712, −15.56565942816637594749342768540, −15.02120898329480019417674318471, −14.51124490609267068181895238046, −14.20904714041741431979615853742, −12.87316257412045570170123041937, −12.81311426566911892789031311059, −11.72243564776574288919111744591, −11.61412770937499088820529268108, −10.61822029052000652311156728651, −10.32607613215945308685700334377, −9.052069465832944814813863569504, −8.403840569983277239534394947763, −7.80051601622674967563616824760, −7.34677056584845444896322601026, −5.93803940801848976957527409700, −5.27306770539016633810253170969, −4.04092755146920523894353866314, −3.03859835267122871133236527477,
3.03859835267122871133236527477, 4.04092755146920523894353866314, 5.27306770539016633810253170969, 5.93803940801848976957527409700, 7.34677056584845444896322601026, 7.80051601622674967563616824760, 8.403840569983277239534394947763, 9.052069465832944814813863569504, 10.32607613215945308685700334377, 10.61822029052000652311156728651, 11.61412770937499088820529268108, 11.72243564776574288919111744591, 12.81311426566911892789031311059, 12.87316257412045570170123041937, 14.20904714041741431979615853742, 14.51124490609267068181895238046, 15.02120898329480019417674318471, 15.56565942816637594749342768540, 16.52017147857990237778726022712, 16.53794698895738274335983513303
