| L(s) = 1 | + 3-s − 3·5-s + 9-s − 3·11-s − 13-s − 3·15-s + 7·17-s − 7·19-s − 9·23-s + 4·25-s + 27-s + 29-s − 2·31-s − 3·33-s + 5·37-s − 39-s + 7·43-s − 3·45-s + 12·47-s + 7·51-s − 6·53-s + 9·55-s − 7·57-s − 2·59-s + 7·61-s + 3·65-s + 8·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.34·5-s + 1/3·9-s − 0.904·11-s − 0.277·13-s − 0.774·15-s + 1.69·17-s − 1.60·19-s − 1.87·23-s + 4/5·25-s + 0.192·27-s + 0.185·29-s − 0.359·31-s − 0.522·33-s + 0.821·37-s − 0.160·39-s + 1.06·43-s − 0.447·45-s + 1.75·47-s + 0.980·51-s − 0.824·53-s + 1.21·55-s − 0.927·57-s − 0.260·59-s + 0.896·61-s + 0.372·65-s + 0.977·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 122304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 17 | \( 1 - 7 T + p T^{2} \) | 1.17.ah |
| 19 | \( 1 + 7 T + p T^{2} \) | 1.19.h |
| 23 | \( 1 + 9 T + p T^{2} \) | 1.23.j |
| 29 | \( 1 - T + p T^{2} \) | 1.29.ab |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 - 5 T + p T^{2} \) | 1.37.af |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 7 T + p T^{2} \) | 1.43.ah |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 2 T + p T^{2} \) | 1.59.c |
| 61 | \( 1 - 7 T + p T^{2} \) | 1.61.ah |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + 2 T + p T^{2} \) | 1.71.c |
| 73 | \( 1 + 7 T + p T^{2} \) | 1.73.h |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 4 T + p T^{2} \) | 1.89.ae |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
| 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
−13.95666484731096, −13.19110130458958, −12.73174036663912, −12.31208761950674, −12.05929435809734, −11.37750018095986, −10.87323261195383, −10.35543967637643, −9.956841111390849, −9.466500920433742, −8.581816373624244, −8.371508646186162, −7.874790554863242, −7.464593574885094, −7.191277620371694, −6.167146828001144, −5.861260730705401, −5.147851197915210, −4.380473037815933, −4.053060947552056, −3.635805653024104, −2.832759709000341, −2.435924434622545, −1.661865052468723, −0.6967910878727961, 0,
0.6967910878727961, 1.661865052468723, 2.435924434622545, 2.832759709000341, 3.635805653024104, 4.053060947552056, 4.380473037815933, 5.147851197915210, 5.861260730705401, 6.167146828001144, 7.191277620371694, 7.464593574885094, 7.874790554863242, 8.371508646186162, 8.581816373624244, 9.466500920433742, 9.956841111390849, 10.35543967637643, 10.87323261195383, 11.37750018095986, 12.05929435809734, 12.31208761950674, 12.73174036663912, 13.19110130458958, 13.95666484731096
