| L(s) = 1 | + 5-s − 7-s + 6·11-s + 4·13-s − 3·17-s + 2·19-s − 2·23-s − 4·25-s − 6·29-s + 4·31-s − 35-s + 5·37-s − 5·41-s + 9·43-s + 5·47-s + 49-s + 6·53-s + 6·55-s − 7·59-s + 14·61-s + 4·65-s + 12·67-s + 8·71-s − 10·73-s − 6·77-s + 5·79-s + 11·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.377·7-s + 1.80·11-s + 1.10·13-s − 0.727·17-s + 0.458·19-s − 0.417·23-s − 4/5·25-s − 1.11·29-s + 0.718·31-s − 0.169·35-s + 0.821·37-s − 0.780·41-s + 1.37·43-s + 0.729·47-s + 1/7·49-s + 0.824·53-s + 0.809·55-s − 0.911·59-s + 1.79·61-s + 0.496·65-s + 1.46·67-s + 0.949·71-s − 1.17·73-s − 0.683·77-s + 0.562·79-s + 1.20·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.023414905\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.023414905\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 3 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| good | 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 5 T + p T^{2} \) | 1.37.af |
| 41 | \( 1 + 5 T + p T^{2} \) | 1.41.f |
| 43 | \( 1 - 9 T + p T^{2} \) | 1.43.aj |
| 47 | \( 1 - 5 T + p T^{2} \) | 1.47.af |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 7 T + p T^{2} \) | 1.59.h |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 5 T + p T^{2} \) | 1.79.af |
| 83 | \( 1 - 11 T + p T^{2} \) | 1.83.al |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
| 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
−9.340295531721786066982181793852, −8.925611791185699598339525240540, −7.925084653820302062303493823668, −6.82650045048228760008620763027, −6.27396856178819027741489400403, −5.56139040449244509480021048832, −4.14342441188947212741921122034, −3.67159395963938027928260839556, −2.23010816328980554422484671986, −1.09138166494557081590095232257,
1.09138166494557081590095232257, 2.23010816328980554422484671986, 3.67159395963938027928260839556, 4.14342441188947212741921122034, 5.56139040449244509480021048832, 6.27396856178819027741489400403, 6.82650045048228760008620763027, 7.925084653820302062303493823668, 8.925611791185699598339525240540, 9.340295531721786066982181793852
