| L(s) = 1 | − 2-s + 3-s + 2·4-s − 2·5-s − 6-s − 2·7-s − 5·8-s + 2·10-s + 2·11-s + 2·12-s − 7·13-s + 2·14-s − 2·15-s + 5·16-s + 7·17-s + 6·19-s − 4·20-s − 2·21-s − 2·22-s + 6·23-s − 5·24-s − 7·25-s + 7·26-s − 27-s − 4·28-s + 29-s + 2·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 4-s − 0.894·5-s − 0.408·6-s − 0.755·7-s − 1.76·8-s + 0.632·10-s + 0.603·11-s + 0.577·12-s − 1.94·13-s + 0.534·14-s − 0.516·15-s + 5/4·16-s + 1.69·17-s + 1.37·19-s − 0.894·20-s − 0.436·21-s − 0.426·22-s + 1.25·23-s − 1.02·24-s − 7/5·25-s + 1.37·26-s − 0.192·27-s − 0.755·28-s + 0.185·29-s + 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4670246137\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4670246137\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 13 | $C_2$ | \( 1 + 7 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + T - T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 7 T + 32 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 6 T + 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - T - 28 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 9 T + 40 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 6 T - 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 2 T - 63 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 6 T - 35 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 14 T + 107 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 2 T - 93 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
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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
−16.67964485929451701471305897548, −15.84408544761433403349579987483, −15.31436868569418472237139205121, −15.27243809376413943797361286045, −14.25352282411028397136124411118, −14.02499101548283092459780281457, −12.73289175444341950296811916307, −12.14357937143744666454386491571, −11.86794614489910264294294396928, −11.42867064720421448825118723410, −10.08439462454078301203589831316, −9.791768698484293397611172234114, −9.254398779241532270121720047097, −8.363264865171006164651980684627, −7.54477102164470802446718447887, −7.22047791579711791825933304934, −6.24892583756641342435496526807, −5.20432204657102172725166254858, −3.47426894868114625087598578954, −2.86393153909535063400258284725,
2.86393153909535063400258284725, 3.47426894868114625087598578954, 5.20432204657102172725166254858, 6.24892583756641342435496526807, 7.22047791579711791825933304934, 7.54477102164470802446718447887, 8.363264865171006164651980684627, 9.254398779241532270121720047097, 9.791768698484293397611172234114, 10.08439462454078301203589831316, 11.42867064720421448825118723410, 11.86794614489910264294294396928, 12.14357937143744666454386491571, 12.73289175444341950296811916307, 14.02499101548283092459780281457, 14.25352282411028397136124411118, 15.27243809376413943797361286045, 15.31436868569418472237139205121, 15.84408544761433403349579987483, 16.67964485929451701471305897548
