| L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s + 8-s − 2·9-s − 10-s + 11-s − 12-s + 13-s + 15-s + 16-s − 17-s − 2·18-s + 6·19-s − 20-s + 22-s − 8·23-s − 24-s + 25-s + 26-s + 5·27-s + 4·29-s + 30-s + 32-s − 33-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 0.353·8-s − 2/3·9-s − 0.316·10-s + 0.301·11-s − 0.288·12-s + 0.277·13-s + 0.258·15-s + 1/4·16-s − 0.242·17-s − 0.471·18-s + 1.37·19-s − 0.223·20-s + 0.213·22-s − 1.66·23-s − 0.204·24-s + 1/5·25-s + 0.196·26-s + 0.962·27-s + 0.742·29-s + 0.182·30-s + 0.176·32-s − 0.174·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.051651003\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.051651003\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 5 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 17 | \( 1 + T \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 11 | \( 1 - T + p T^{2} \) | 1.11.ab |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + 7 T + p T^{2} \) | 1.53.h |
| 59 | \( 1 + 2 T + p T^{2} \) | 1.59.c |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 - 3 T + p T^{2} \) | 1.71.ad |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 - 15 T + p T^{2} \) | 1.79.ap |
| 83 | \( 1 + 2 T + p T^{2} \) | 1.83.c |
| 89 | \( 1 + 3 T + p T^{2} \) | 1.89.d |
| 97 | \( 1 + p T^{2} \) | 1.97.a |
| 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
−7.74876034261526622280953460143, −6.92402314995680862678687926100, −6.29507868438233452700540640013, −5.68275396203538919628256641423, −5.08387292226239265768758949158, −4.30596646715780181078229358578, −3.55810032163963275833456571533, −2.88785158995212483124678917167, −1.82250017882535524231054396707, −0.64433564220830929019379645591,
0.64433564220830929019379645591, 1.82250017882535524231054396707, 2.88785158995212483124678917167, 3.55810032163963275833456571533, 4.30596646715780181078229358578, 5.08387292226239265768758949158, 5.68275396203538919628256641423, 6.29507868438233452700540640013, 6.92402314995680862678687926100, 7.74876034261526622280953460143
