| L(s) = 1 | + 3-s + 5-s + 3·9-s − 8·11-s + 13-s + 15-s − 3·17-s − 8·19-s − 5·23-s + 5·25-s + 8·27-s − 7·29-s + 8·31-s − 8·33-s + 20·37-s + 39-s + 5·41-s + 5·43-s + 3·45-s + 7·47-s − 14·49-s − 3·51-s − 11·53-s − 8·55-s − 8·57-s − 3·59-s − 11·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 9-s − 2.41·11-s + 0.277·13-s + 0.258·15-s − 0.727·17-s − 1.83·19-s − 1.04·23-s + 25-s + 1.53·27-s − 1.29·29-s + 1.43·31-s − 1.39·33-s + 3.28·37-s + 0.160·39-s + 0.780·41-s + 0.762·43-s + 0.447·45-s + 1.02·47-s − 2·49-s − 0.420·51-s − 1.51·53-s − 1.07·55-s − 1.05·57-s − 0.390·59-s − 1.40·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5776 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.019358230\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.019358230\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_2$ | \( 1 + 8 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - T - 12 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 5 T + 2 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 7 T + 20 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 5 T - 16 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 7 T + 2 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 11 T + 68 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 3 T - 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 11 T + 60 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 3 T - 58 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 11 T + 50 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 15 T + 152 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 3 T - 80 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 19 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−14.76586366114238299366274732007, −14.29974575719456045326622773158, −13.54349859961333264765282700419, −13.11069918964448992657991348226, −12.74655459434758243945139301072, −12.56425310513379079331226375170, −11.07130322013113521553593114138, −11.06774901692491150864147072101, −10.13794998726485266646247897559, −10.03822225854232995061329335650, −9.083872573304780836458674175901, −8.523628327780524062372942515050, −7.68782966426915268740231684369, −7.65381165678650911206172693551, −6.24816901613009100736364334993, −6.09883239151249176774240439136, −4.60432761726607850681992359706, −4.53039496134745697616897108822, −2.88959967239244464973068077936, −2.23275176588073520095478123304,
2.23275176588073520095478123304, 2.88959967239244464973068077936, 4.53039496134745697616897108822, 4.60432761726607850681992359706, 6.09883239151249176774240439136, 6.24816901613009100736364334993, 7.65381165678650911206172693551, 7.68782966426915268740231684369, 8.523628327780524062372942515050, 9.083872573304780836458674175901, 10.03822225854232995061329335650, 10.13794998726485266646247897559, 11.06774901692491150864147072101, 11.07130322013113521553593114138, 12.56425310513379079331226375170, 12.74655459434758243945139301072, 13.11069918964448992657991348226, 13.54349859961333264765282700419, 14.29974575719456045326622773158, 14.76586366114238299366274732007
