| L(s) = 1 | + 7-s − 3·9-s + 6·11-s + 2·13-s + 3·17-s + 6·19-s + 23-s + 3·29-s + 3·31-s − 37-s + 9·41-s − 8·43-s + 4·47-s − 6·49-s − 53-s − 59-s + 8·61-s − 3·63-s − 7·67-s + 5·71-s + 6·73-s + 6·77-s + 9·81-s − 11·83-s + 4·89-s + 2·91-s − 6·97-s + ⋯ |
| L(s) = 1 | + 0.377·7-s − 9-s + 1.80·11-s + 0.554·13-s + 0.727·17-s + 1.37·19-s + 0.208·23-s + 0.557·29-s + 0.538·31-s − 0.164·37-s + 1.40·41-s − 1.21·43-s + 0.583·47-s − 6/7·49-s − 0.137·53-s − 0.130·59-s + 1.02·61-s − 0.377·63-s − 0.855·67-s + 0.593·71-s + 0.702·73-s + 0.683·77-s + 81-s − 1.20·83-s + 0.423·89-s + 0.209·91-s − 0.609·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.722071172\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.722071172\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 23 | \( 1 - T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 - 3 T + p T^{2} \) | 1.31.ad |
| 37 | \( 1 + T + p T^{2} \) | 1.37.b |
| 41 | \( 1 - 9 T + p T^{2} \) | 1.41.aj |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 + T + p T^{2} \) | 1.53.b |
| 59 | \( 1 + T + p T^{2} \) | 1.59.b |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 + 7 T + p T^{2} \) | 1.67.h |
| 71 | \( 1 - 5 T + p T^{2} \) | 1.71.af |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 11 T + p T^{2} \) | 1.83.l |
| 89 | \( 1 - 4 T + p T^{2} \) | 1.89.ae |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
| 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
−7.82298979715247855291938725747, −6.94902272641391156528403180744, −6.35281879855327788734414431038, −5.69293563485133760917944078698, −5.04358822214041823989359279593, −4.12028213959060019465577139089, −3.44763919960524874560818059221, −2.76880603813738629274041989585, −1.51929783951366995615555153225, −0.884396337875045552140964420724,
0.884396337875045552140964420724, 1.51929783951366995615555153225, 2.76880603813738629274041989585, 3.44763919960524874560818059221, 4.12028213959060019465577139089, 5.04358822214041823989359279593, 5.69293563485133760917944078698, 6.35281879855327788734414431038, 6.94902272641391156528403180744, 7.82298979715247855291938725747
