| L(s) = 1 | + 3-s + 4·7-s − 2·9-s − 11-s − 13-s − 7·17-s + 3·19-s + 4·21-s − 5·27-s − 4·29-s − 6·31-s − 33-s − 8·37-s − 39-s − 5·41-s + 4·43-s − 12·47-s + 9·49-s − 7·51-s − 10·53-s + 3·57-s − 4·59-s + 8·61-s − 8·63-s + 9·67-s + 8·71-s + 13·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.51·7-s − 2/3·9-s − 0.301·11-s − 0.277·13-s − 1.69·17-s + 0.688·19-s + 0.872·21-s − 0.962·27-s − 0.742·29-s − 1.07·31-s − 0.174·33-s − 1.31·37-s − 0.160·39-s − 0.780·41-s + 0.609·43-s − 1.75·47-s + 9/7·49-s − 0.980·51-s − 1.37·53-s + 0.397·57-s − 0.520·59-s + 1.02·61-s − 1.00·63-s + 1.09·67-s + 0.949·71-s + 1.52·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 17 | \( 1 + 7 T + p T^{2} \) | 1.17.h |
| 19 | \( 1 - 3 T + p T^{2} \) | 1.19.ad |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + 5 T + p T^{2} \) | 1.41.f |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 - 9 T + p T^{2} \) | 1.67.aj |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 13 T + p T^{2} \) | 1.73.an |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 3 T + p T^{2} \) | 1.83.d |
| 89 | \( 1 + 11 T + p T^{2} \) | 1.89.l |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
| 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
−8.036018389954580931453125919175, −7.28543628872390781437514578146, −6.53555126165593338519170331548, −5.35413770874386861402748234427, −5.08009615398371979250513331732, −4.11689548602201027798426673135, −3.24701724289112651672118397483, −2.22653514858673000839577247140, −1.68859007806646180430559011829, 0,
1.68859007806646180430559011829, 2.22653514858673000839577247140, 3.24701724289112651672118397483, 4.11689548602201027798426673135, 5.08009615398371979250513331732, 5.35413770874386861402748234427, 6.53555126165593338519170331548, 7.28543628872390781437514578146, 8.036018389954580931453125919175
