| L(s) = 1 | + 2-s − 3-s − 3·5-s − 6-s + 7-s − 8-s − 3·10-s − 3·11-s − 2·13-s + 14-s + 3·15-s − 16-s − 12·17-s − 2·19-s − 21-s − 3·22-s − 6·23-s + 24-s + 5·25-s − 2·26-s + 27-s + 3·29-s + 3·30-s − 5·31-s + 3·33-s − 12·34-s − 3·35-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s − 1.34·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s − 0.948·10-s − 0.904·11-s − 0.554·13-s + 0.267·14-s + 0.774·15-s − 1/4·16-s − 2.91·17-s − 0.458·19-s − 0.218·21-s − 0.639·22-s − 1.25·23-s + 0.204·24-s + 25-s − 0.392·26-s + 0.192·27-s + 0.557·29-s + 0.547·30-s − 0.898·31-s + 0.522·33-s − 2.05·34-s − 0.507·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 224676 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224676 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4941971074\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4941971074\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | $C_2$ | \( 1 + T + T^{2} \) |
| 79 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 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 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 3 T - 20 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 5 T - 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 2 T - 39 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 12 T + 97 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.37193489759604086516973187303, −10.89580251329802385345498468294, −10.78000242012041130977922932821, −9.750570699806988571764051211845, −9.729005853045753269749969800907, −8.737593859563243842733521864452, −8.359428138228071136730557313109, −8.293092371864962204872256863320, −7.48132354156701232435435380219, −6.99402742478826455858384739460, −6.69655665861316174443047515432, −6.03622054323768741358538125152, −5.46986437557971487932800238194, −4.80221643572788490104596107627, −4.64871906544376044711900227124, −3.94413546107022654083719115278, −3.70597661740085927145734972897, −2.35963452173841381631412012457, −2.34797346546542318001608131624, −0.37805627091306981900321532580,
0.37805627091306981900321532580, 2.34797346546542318001608131624, 2.35963452173841381631412012457, 3.70597661740085927145734972897, 3.94413546107022654083719115278, 4.64871906544376044711900227124, 4.80221643572788490104596107627, 5.46986437557971487932800238194, 6.03622054323768741358538125152, 6.69655665861316174443047515432, 6.99402742478826455858384739460, 7.48132354156701232435435380219, 8.293092371864962204872256863320, 8.359428138228071136730557313109, 8.737593859563243842733521864452, 9.729005853045753269749969800907, 9.750570699806988571764051211845, 10.78000242012041130977922932821, 10.89580251329802385345498468294, 11.37193489759604086516973187303
