| L(s) = 1 | − 3-s − 7-s − 2·9-s + 5·11-s + 5·13-s + 5·17-s + 21-s + 8·23-s + 5·27-s + 29-s − 2·31-s − 5·33-s + 4·37-s − 5·39-s + 2·41-s − 4·43-s + 13·47-s + 49-s − 5·51-s − 8·53-s + 4·59-s + 2·61-s + 2·63-s + 8·67-s − 8·69-s + 12·71-s + 6·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.377·7-s − 2/3·9-s + 1.50·11-s + 1.38·13-s + 1.21·17-s + 0.218·21-s + 1.66·23-s + 0.962·27-s + 0.185·29-s − 0.359·31-s − 0.870·33-s + 0.657·37-s − 0.800·39-s + 0.312·41-s − 0.609·43-s + 1.89·47-s + 1/7·49-s − 0.700·51-s − 1.09·53-s + 0.520·59-s + 0.256·61-s + 0.251·63-s + 0.977·67-s − 0.963·69-s + 1.42·71-s + 0.702·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.163128758\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.163128758\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 11 | \( 1 - 5 T + p T^{2} \) | 1.11.af |
| 13 | \( 1 - 5 T + p T^{2} \) | 1.13.af |
| 17 | \( 1 - 5 T + p T^{2} \) | 1.17.af |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 - T + p T^{2} \) | 1.29.ab |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 13 T + p T^{2} \) | 1.47.an |
| 53 | \( 1 + 8 T + p T^{2} \) | 1.53.i |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 11 T + p T^{2} \) | 1.79.l |
| 83 | \( 1 + 8 T + p T^{2} \) | 1.83.i |
| 89 | \( 1 + 18 T + p T^{2} \) | 1.89.s |
| 97 | \( 1 - 5 T + p T^{2} \) | 1.97.af |
| 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
−16.61298246211103, −16.05902437328708, −15.34324526550738, −14.76322175170429, −14.03404438464019, −13.89298485813228, −12.79289287668442, −12.54957753191482, −11.75149827907419, −11.17455924045462, −11.01313856088257, −10.03459150167856, −9.438782667877033, −8.772355343368943, −8.424382043019265, −7.401725640444158, −6.742994978588663, −6.183036616999127, −5.687677432020664, −4.986401511677521, −3.972831717726541, −3.477084229840239, −2.703426594221983, −1.320436178537670, −0.8221946206743418,
0.8221946206743418, 1.320436178537670, 2.703426594221983, 3.477084229840239, 3.972831717726541, 4.986401511677521, 5.687677432020664, 6.183036616999127, 6.742994978588663, 7.401725640444158, 8.424382043019265, 8.772355343368943, 9.438782667877033, 10.03459150167856, 11.01313856088257, 11.17455924045462, 11.75149827907419, 12.54957753191482, 12.79289287668442, 13.89298485813228, 14.03404438464019, 14.76322175170429, 15.34324526550738, 16.05902437328708, 16.61298246211103
