| L(s) = 1 | + 3-s − 7-s + 9-s − 11-s − 4·13-s + 3·17-s + 5·19-s − 21-s + 3·23-s + 27-s − 6·29-s + 8·31-s − 33-s − 7·37-s − 4·39-s + 9·41-s + 8·43-s − 3·47-s − 6·49-s + 3·51-s + 6·53-s + 5·57-s + 3·59-s + 14·61-s − 63-s + 2·67-s + 3·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.377·7-s + 1/3·9-s − 0.301·11-s − 1.10·13-s + 0.727·17-s + 1.14·19-s − 0.218·21-s + 0.625·23-s + 0.192·27-s − 1.11·29-s + 1.43·31-s − 0.174·33-s − 1.15·37-s − 0.640·39-s + 1.40·41-s + 1.21·43-s − 0.437·47-s − 6/7·49-s + 0.420·51-s + 0.824·53-s + 0.662·57-s + 0.390·59-s + 1.79·61-s − 0.125·63-s + 0.244·67-s + 0.361·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.153881380\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.153881380\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 - 9 T + p T^{2} \) | 1.41.aj |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 3 T + p T^{2} \) | 1.59.ad |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 + 9 T + p T^{2} \) | 1.71.j |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + T + p T^{2} \) | 1.79.b |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 18 T + p T^{2} \) | 1.89.as |
| 97 | \( 1 - 11 T + p T^{2} \) | 1.97.al |
| 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
−8.696143167997132278850002282763, −7.65483689365950253173228694258, −7.45271645794781968966571895593, −6.49005999941517119332358636278, −5.48077983091559049805038341357, −4.87166627193647095530866849012, −3.78274651895588828002510294819, −3.00933788925534646274507562145, −2.23498748075493272086475758078, −0.855913662863658945601921871321,
0.855913662863658945601921871321, 2.23498748075493272086475758078, 3.00933788925534646274507562145, 3.78274651895588828002510294819, 4.87166627193647095530866849012, 5.48077983091559049805038341357, 6.49005999941517119332358636278, 7.45271645794781968966571895593, 7.65483689365950253173228694258, 8.696143167997132278850002282763
