| L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s − 2·7-s + 8-s + 9-s − 10-s + 11-s + 12-s − 2·14-s − 15-s + 16-s + 6·17-s + 18-s + 4·19-s − 20-s − 2·21-s + 22-s + 24-s + 25-s + 27-s − 2·28-s − 30-s − 2·31-s + 32-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 + 0.301·11-s + 0.288·12-s − 0.534·14-s − 0.258·15-s + 1/4·16-s + 1.45·17-s + 0.235·18-s + 0.917·19-s − 0.223·20-s − 0.436·21-s + 0.213·22-s + 0.204·24-s + 1/5·25-s + 0.192·27-s − 0.377·28-s − 0.182·30-s − 0.359·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.911438598\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.911438598\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 11 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 16 T + p T^{2} \) | 1.97.aq |
| 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
−14.44139867351052, −13.88547700362388, −13.43900674712795, −12.91894602697948, −12.34185151975424, −12.04547381159135, −11.51715765517040, −10.86505925815301, −10.18689228148158, −9.859908116172132, −9.204215687635230, −8.761483684269222, −7.893624567032694, −7.582249655046620, −7.117398159189260, −6.382367399997987, −5.897240323734436, −5.254278448601842, −4.657240468617697, −3.869540352783071, −3.471874355154971, −3.039121492621753, −2.336307135713256, −1.407384729929827, −0.6852012423304386,
0.6852012423304386, 1.407384729929827, 2.336307135713256, 3.039121492621753, 3.471874355154971, 3.869540352783071, 4.657240468617697, 5.254278448601842, 5.897240323734436, 6.382367399997987, 7.117398159189260, 7.582249655046620, 7.893624567032694, 8.761483684269222, 9.204215687635230, 9.859908116172132, 10.18689228148158, 10.86505925815301, 11.51715765517040, 12.04547381159135, 12.34185151975424, 12.91894602697948, 13.43900674712795, 13.88547700362388, 14.44139867351052
