| L(s) = 1 | − 3-s + 9-s − 7·13-s − 5·17-s − 2·19-s − 3·23-s − 27-s − 3·29-s + 9·31-s + 8·37-s + 7·39-s + 3·41-s + 43-s + 8·47-s + 5·51-s − 3·53-s + 2·57-s + 7·59-s − 61-s − 12·67-s + 3·69-s + 12·71-s + 4·73-s + 12·79-s + 81-s − 3·83-s + 3·87-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 1.94·13-s − 1.21·17-s − 0.458·19-s − 0.625·23-s − 0.192·27-s − 0.557·29-s + 1.61·31-s + 1.31·37-s + 1.12·39-s + 0.468·41-s + 0.152·43-s + 1.16·47-s + 0.700·51-s − 0.412·53-s + 0.264·57-s + 0.911·59-s − 0.128·61-s − 1.46·67-s + 0.361·69-s + 1.42·71-s + 0.468·73-s + 1.35·79-s + 1/9·81-s − 0.329·83-s + 0.321·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.165739167\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.165739167\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| good | 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 7 T + p T^{2} \) | 1.13.h |
| 17 | \( 1 + 5 T + p T^{2} \) | 1.17.f |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 - 9 T + p T^{2} \) | 1.31.aj |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 3 T + p T^{2} \) | 1.53.d |
| 59 | \( 1 - 7 T + p T^{2} \) | 1.59.ah |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 + 3 T + p T^{2} \) | 1.83.d |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
| 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
−12.74256233739251, −12.51426678441730, −11.87425692566948, −11.65061979117699, −11.08693033574003, −10.57183159180012, −10.14433036072380, −9.702307235742278, −9.269159936602568, −8.771111312842198, −8.094485795447186, −7.604151577084440, −7.306146069193391, −6.570393057106760, −6.322160814465306, −5.749895010053456, −5.078920860363772, −4.646878298347539, −4.335229303170569, −3.717146470764434, −2.810788486783087, −2.310367397986301, −2.031821163654479, −0.9298603082859630, −0.3630736253559480,
0.3630736253559480, 0.9298603082859630, 2.031821163654479, 2.310367397986301, 2.810788486783087, 3.717146470764434, 4.335229303170569, 4.646878298347539, 5.078920860363772, 5.749895010053456, 6.322160814465306, 6.570393057106760, 7.306146069193391, 7.604151577084440, 8.094485795447186, 8.771111312842198, 9.269159936602568, 9.702307235742278, 10.14433036072380, 10.57183159180012, 11.08693033574003, 11.65061979117699, 11.87425692566948, 12.51426678441730, 12.74256233739251
