| L(s) = 1 | − 2·3-s − 96·5-s + 243·9-s + 720·11-s − 1.14e3·13-s + 192·15-s + 1.25e3·17-s − 94·19-s − 96·23-s + 3.12e3·25-s − 1.45e3·27-s − 8.74e3·29-s − 6.24e3·31-s − 1.44e3·33-s + 1.07e4·37-s + 2.28e3·39-s − 2.40e4·41-s − 1.83e4·43-s − 2.33e4·45-s − 2.58e4·47-s − 2.50e3·51-s − 1.01e3·53-s − 6.91e4·55-s + 188·57-s + 1.24e3·59-s + 7.59e3·61-s + 1.09e5·65-s + ⋯ |
| L(s) = 1 | − 0.128·3-s − 1.71·5-s + 9-s + 1.79·11-s − 1.87·13-s + 0.220·15-s + 1.05·17-s − 0.0597·19-s − 0.0378·23-s + 25-s − 0.382·27-s − 1.93·29-s − 1.16·31-s − 0.230·33-s + 1.29·37-s + 0.240·39-s − 2.23·41-s − 1.51·43-s − 1.71·45-s − 1.70·47-s − 0.135·51-s − 0.0495·53-s − 3.08·55-s + 0.00766·57-s + 0.0464·59-s + 0.261·61-s + 3.22·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38416 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.2481889199\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2481889199\) |
| \(L(\frac{7}{2})\) |
|
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 2 T - 239 T^{2} + 2 p^{5} T^{3} + p^{10} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 96 T + 6091 T^{2} + 96 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 720 T + 357349 T^{2} - 720 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 44 p T + p^{5} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 1254 T + 152659 T^{2} - 1254 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 94 T - 2467263 T^{2} + 94 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 96 T - 6427127 T^{2} + 96 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 4374 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 6244 T + 10358385 T^{2} + 6244 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 10798 T + 47252847 T^{2} - 10798 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 12006 T + p^{5} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 9160 T + p^{5} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 25836 T + 438153889 T^{2} + 25836 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 1014 T - 417167297 T^{2} + 1014 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 1242 T - 713381735 T^{2} - 1242 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 7592 T - 786957837 T^{2} - 7592 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 41132 T + 341716317 T^{2} + 41132 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 37632 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 13438 T - 1892491749 T^{2} + 13438 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 6248 T - 3038018895 T^{2} + 6248 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 25254 T + p^{5} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 45126 T - 3547703573 T^{2} + 45126 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 107222 T + p^{5} T^{2} )^{2} \) |
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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
−12.09420721988347736681143286235, −11.42443260354620065352516653537, −11.27787190625226604686068967977, −10.12257303442602635798676766561, −9.992983832022863350433335233121, −9.454546502938926756392921705470, −8.909914112048358104686662416432, −8.190046702839338150990478563924, −7.67308618641081407450477127771, −7.14244037120620312402401737135, −7.13502873830267578190961981508, −6.24092906208544392485915590777, −5.43019229997453720803397108305, −4.67261834996503463971328498347, −4.32530449232325762879222275988, −3.46137141073236979985375330950, −3.44209387921446104811568526060, −1.90407419139002326756593550545, −1.37537934895147061095476652506, −0.16054087529627275685443708536,
0.16054087529627275685443708536, 1.37537934895147061095476652506, 1.90407419139002326756593550545, 3.44209387921446104811568526060, 3.46137141073236979985375330950, 4.32530449232325762879222275988, 4.67261834996503463971328498347, 5.43019229997453720803397108305, 6.24092906208544392485915590777, 7.13502873830267578190961981508, 7.14244037120620312402401737135, 7.67308618641081407450477127771, 8.190046702839338150990478563924, 8.909914112048358104686662416432, 9.454546502938926756392921705470, 9.992983832022863350433335233121, 10.12257303442602635798676766561, 11.27787190625226604686068967977, 11.42443260354620065352516653537, 12.09420721988347736681143286235
