| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 8-s + 9-s − 11-s − 12-s + 13-s + 16-s + 3·17-s − 18-s − 7·19-s + 22-s + 4·23-s + 24-s − 26-s − 27-s + 2·29-s + 3·31-s − 32-s + 33-s − 3·34-s + 36-s − 2·37-s + 7·38-s − 39-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.301·11-s − 0.288·12-s + 0.277·13-s + 1/4·16-s + 0.727·17-s − 0.235·18-s − 1.60·19-s + 0.213·22-s + 0.834·23-s + 0.204·24-s − 0.196·26-s − 0.192·27-s + 0.371·29-s + 0.538·31-s − 0.176·32-s + 0.174·33-s − 0.514·34-s + 1/6·36-s − 0.328·37-s + 1.13·38-s − 0.160·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ |
|---|
| bad | 2 | \( 1 + T \) | |
| 3 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 + 7 T + p T^{2} \) | 1.19.h |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 3 T + p T^{2} \) | 1.31.ad |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - T + p T^{2} \) | 1.61.ab |
| 67 | \( 1 - 11 T + p T^{2} \) | 1.67.al |
| 71 | \( 1 + T + p T^{2} \) | 1.71.b |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 + 7 T + p T^{2} \) | 1.83.h |
| 89 | \( 1 + 18 T + p T^{2} \) | 1.89.s |
| 97 | \( 1 - 12 T + p T^{2} \) | 1.97.am |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.31051793253173, −13.32883331131997, −12.97702813256050, −12.57143591236291, −12.02008575113010, −11.38921205774272, −11.15661991359324, −10.49340053364289, −10.15460874029904, −9.762576622392214, −8.940050778462383, −8.634749035406994, −8.083267697695306, −7.541482561964946, −6.974119418917588, −6.405245956911110, −6.100569167265777, −5.351657969805176, −4.865327517281081, −4.215200486355545, −3.534429220033395, −2.845724361361119, −2.196573157217473, −1.462916921877241, −0.7935327214271474, 0,
0.7935327214271474, 1.462916921877241, 2.196573157217473, 2.845724361361119, 3.534429220033395, 4.215200486355545, 4.865327517281081, 5.351657969805176, 6.100569167265777, 6.405245956911110, 6.974119418917588, 7.541482561964946, 8.083267697695306, 8.634749035406994, 8.940050778462383, 9.762576622392214, 10.15460874029904, 10.49340053364289, 11.15661991359324, 11.38921205774272, 12.02008575113010, 12.57143591236291, 12.97702813256050, 13.32883331131997, 14.31051793253173
