| L(s) = 1 | − 2-s − 3·3-s + 4-s + 3·6-s − 8-s + 6·9-s − 5·11-s − 3·12-s + 13-s + 16-s + 4·17-s − 6·18-s − 2·19-s + 5·22-s + 5·23-s + 3·24-s − 5·25-s − 26-s − 9·27-s + 4·29-s − 31-s − 32-s + 15·33-s − 4·34-s + 6·36-s + 7·37-s + 2·38-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.73·3-s + 1/2·4-s + 1.22·6-s − 0.353·8-s + 2·9-s − 1.50·11-s − 0.866·12-s + 0.277·13-s + 1/4·16-s + 0.970·17-s − 1.41·18-s − 0.458·19-s + 1.06·22-s + 1.04·23-s + 0.612·24-s − 25-s − 0.196·26-s − 1.73·27-s + 0.742·29-s − 0.179·31-s − 0.176·32-s + 2.61·33-s − 0.685·34-s + 36-s + 1.15·37-s + 0.324·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1274 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1274 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 3 | \( 1 + p T + p T^{2} \) | 1.3.d |
| 5 | \( 1 + p T^{2} \) | 1.5.a |
| 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 5 T + p T^{2} \) | 1.23.af |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 + T + p T^{2} \) | 1.31.b |
| 37 | \( 1 - 7 T + p T^{2} \) | 1.37.ah |
| 41 | \( 1 - 9 T + p T^{2} \) | 1.41.aj |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 - 7 T + p T^{2} \) | 1.47.ah |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 13 T + p T^{2} \) | 1.61.n |
| 67 | \( 1 - 11 T + p T^{2} \) | 1.67.al |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 7 T + p T^{2} \) | 1.73.h |
| 79 | \( 1 + 17 T + p T^{2} \) | 1.79.r |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 + 5 T + p T^{2} \) | 1.97.f |
| 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
−9.612162736352015844529352631345, −8.312051351400959597402385747456, −7.58448688447778853120023876132, −6.77348571516541885573598245021, −5.83826962229613740807236396926, −5.36198914955171371154112152465, −4.33501878167790135292580228215, −2.78400872191447728073145525196, −1.23571105217925524886590997058, 0,
1.23571105217925524886590997058, 2.78400872191447728073145525196, 4.33501878167790135292580228215, 5.36198914955171371154112152465, 5.83826962229613740807236396926, 6.77348571516541885573598245021, 7.58448688447778853120023876132, 8.312051351400959597402385747456, 9.612162736352015844529352631345
