| L(s) = 1 | + 2-s + 3-s + 4-s + 5-s + 6-s + 2·7-s + 8-s + 9-s + 10-s − 3·11-s + 12-s + 2·14-s + 15-s + 16-s − 3·17-s + 18-s − 4·19-s + 20-s + 2·21-s − 3·22-s − 3·23-s + 24-s − 4·25-s + 27-s + 2·28-s + 3·29-s + 30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 0.755·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.904·11-s + 0.288·12-s + 0.534·14-s + 0.258·15-s + 1/4·16-s − 0.727·17-s + 0.235·18-s − 0.917·19-s + 0.223·20-s + 0.436·21-s − 0.639·22-s − 0.625·23-s + 0.204·24-s − 4/5·25-s + 0.192·27-s + 0.377·28-s + 0.557·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 41574 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41574 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| 41 | \( 1 + T \) | |
| good | 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 - T + p T^{2} \) | 1.31.ab |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 - 11 T + p T^{2} \) | 1.53.al |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 - T + p T^{2} \) | 1.79.ab |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 3 T + p T^{2} \) | 1.97.d |
| 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
−14.99319555777168, −14.52370730825084, −13.79237140174287, −13.50439711654045, −13.18366726491282, −12.50826277037608, −11.92998095568114, −11.44800312718140, −10.77739920731399, −10.33250454045980, −9.902132092181019, −9.100268621689621, −8.550197223288622, −7.948575026326338, −7.697513397882098, −6.752477224484815, −6.387604411109730, −5.627352894036658, −5.102826096524948, −4.469071837618588, −4.027511690449637, −3.206026663824517, −2.380457562670106, −2.138467806242749, −1.322651292839183, 0,
1.322651292839183, 2.138467806242749, 2.380457562670106, 3.206026663824517, 4.027511690449637, 4.469071837618588, 5.102826096524948, 5.627352894036658, 6.387604411109730, 6.752477224484815, 7.697513397882098, 7.948575026326338, 8.550197223288622, 9.100268621689621, 9.902132092181019, 10.33250454045980, 10.77739920731399, 11.44800312718140, 11.92998095568114, 12.50826277037608, 13.18366726491282, 13.50439711654045, 13.79237140174287, 14.52370730825084, 14.99319555777168
