| L(s) = 1 | + 2-s − 3·3-s − 4-s − 3·6-s + 3·7-s − 3·8-s + 6·9-s + 3·12-s + 5·13-s + 3·14-s − 16-s − 17-s + 6·18-s + 4·19-s − 9·21-s − 4·23-s + 9·24-s + 5·26-s − 9·27-s − 3·28-s + 31-s + 5·32-s − 34-s − 6·36-s + 2·37-s + 4·38-s − 15·39-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.73·3-s − 1/2·4-s − 1.22·6-s + 1.13·7-s − 1.06·8-s + 2·9-s + 0.866·12-s + 1.38·13-s + 0.801·14-s − 1/4·16-s − 0.242·17-s + 1.41·18-s + 0.917·19-s − 1.96·21-s − 0.834·23-s + 1.83·24-s + 0.980·26-s − 1.73·27-s − 0.566·28-s + 0.179·31-s + 0.883·32-s − 0.171·34-s − 36-s + 0.328·37-s + 0.648·38-s − 2.40·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.418974069\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.418974069\) |
| \(L(\frac{3}{2})\) |
|
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 | 5 | \( 1 \) | |
| 11 | \( 1 \) | |
| 17 | \( 1 + T \) | |
| good | 2 | \( 1 - T + p T^{2} \) | 1.2.ab |
| 3 | \( 1 + p T + p T^{2} \) | 1.3.d |
| 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 13 | \( 1 - 5 T + p T^{2} \) | 1.13.af |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - T + p T^{2} \) | 1.31.ab |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + 7 T + p T^{2} \) | 1.53.h |
| 59 | \( 1 + 2 T + p T^{2} \) | 1.59.c |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 + 13 T + p T^{2} \) | 1.71.n |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 - 13 T + p T^{2} \) | 1.79.an |
| 83 | \( 1 - 14 T + p T^{2} \) | 1.83.ao |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.32806286478392, −13.89759212335364, −13.40321397295253, −13.01052888734850, −12.26071085488812, −11.92010298526310, −11.55554725970331, −10.99073374774590, −10.69337254172779, −9.904131035821986, −9.472192927151775, −8.672289992222304, −8.125471062531594, −7.663503668562131, −6.568923426058430, −6.479419323141182, −5.672711207823028, −5.381046416378157, −4.799777790388569, −4.404303682309293, −3.756663762149252, −3.114660287576590, −1.817043887599905, −1.268503064843786, −0.4621016390791855,
0.4621016390791855, 1.268503064843786, 1.817043887599905, 3.114660287576590, 3.756663762149252, 4.404303682309293, 4.799777790388569, 5.381046416378157, 5.672711207823028, 6.479419323141182, 6.568923426058430, 7.663503668562131, 8.125471062531594, 8.672289992222304, 9.472192927151775, 9.904131035821986, 10.69337254172779, 10.99073374774590, 11.55554725970331, 11.92010298526310, 12.26071085488812, 13.01052888734850, 13.40321397295253, 13.89759212335364, 14.32806286478392
