| L(s) = 1 | − 3·2-s − 3·3-s + 8·4-s − 18·5-s + 9·6-s − 2·7-s − 45·8-s + 54·10-s − 30·11-s − 24·12-s + 65·13-s + 6·14-s + 54·15-s + 135·16-s + 111·17-s + 46·19-s − 144·20-s + 6·21-s + 90·22-s + 6·23-s + 135·24-s − 7·25-s − 195·26-s + 27·27-s − 16·28-s + 105·29-s − 162·30-s + ⋯ |
| L(s) = 1 | − 1.06·2-s − 0.577·3-s + 4-s − 1.60·5-s + 0.612·6-s − 0.107·7-s − 1.98·8-s + 1.70·10-s − 0.822·11-s − 0.577·12-s + 1.38·13-s + 0.114·14-s + 0.929·15-s + 2.10·16-s + 1.58·17-s + 0.555·19-s − 1.60·20-s + 0.0623·21-s + 0.872·22-s + 0.0543·23-s + 1.14·24-s − 0.0559·25-s − 1.47·26-s + 0.192·27-s − 0.107·28-s + 0.672·29-s − 0.985·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3996922168\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3996922168\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| 13 | $C_2$ | \( 1 - 5 p T + p^{3} T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + 3 T + T^{2} + 3 p^{3} T^{3} + p^{6} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + 9 T + p^{3} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 2 T - 339 T^{2} + 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 30 T - 431 T^{2} + 30 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 111 T + 7408 T^{2} - 111 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 46 T - 4743 T^{2} - 46 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 6 T - 12131 T^{2} - 6 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 105 T - 13364 T^{2} - 105 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 100 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 17 T - 50364 T^{2} + 17 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 231 T - 15560 T^{2} - 231 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 514 T + 184689 T^{2} - 514 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 162 T + p^{3} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 639 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 600 T + 154621 T^{2} + 600 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 233 T - 172692 T^{2} + 233 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 926 T + 556713 T^{2} + 926 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 930 T + 506989 T^{2} - 930 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 253 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 1324 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 810 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 498 T - 456965 T^{2} + 498 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 5 p T + p^{3} T^{2} )( 1 + 19 p T + p^{3} T^{2} ) \) |
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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
−16.24969277513667795997208849911, −15.50821788485238729494204240858, −15.35502232383205681484682292362, −14.54885741006108173524894542380, −13.72168642900760615102413011191, −12.58044740512525358456738198264, −12.31903367622654020636877500208, −11.48386637473184268745797676856, −11.45846709897553172883995877490, −10.53413358628060379033607761988, −9.978461626752906803190522287028, −8.949347206956979154885561145505, −8.515736016715523211199606996911, −7.55338517681390666667414049695, −7.44163828027592214396493997798, −5.95406759271731495737716299802, −5.69451087536718866451308193320, −3.89582735149094349531468270883, −3.07740421174496972295053312568, −0.66173629771368750311017618197,
0.66173629771368750311017618197, 3.07740421174496972295053312568, 3.89582735149094349531468270883, 5.69451087536718866451308193320, 5.95406759271731495737716299802, 7.44163828027592214396493997798, 7.55338517681390666667414049695, 8.515736016715523211199606996911, 8.949347206956979154885561145505, 9.978461626752906803190522287028, 10.53413358628060379033607761988, 11.45846709897553172883995877490, 11.48386637473184268745797676856, 12.31903367622654020636877500208, 12.58044740512525358456738198264, 13.72168642900760615102413011191, 14.54885741006108173524894542380, 15.35502232383205681484682292362, 15.50821788485238729494204240858, 16.24969277513667795997208849911
