| L(s) = 1 | − 3-s − 2·7-s + 9-s − 4·11-s + 13-s + 6·17-s − 5·19-s + 2·21-s + 6·23-s − 27-s + 29-s + 10·31-s + 4·33-s + 3·37-s − 39-s − 7·41-s + 8·43-s + 7·47-s − 3·49-s − 6·51-s − 3·53-s + 5·57-s + 6·59-s − 2·61-s − 2·63-s + 9·67-s − 6·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.755·7-s + 1/3·9-s − 1.20·11-s + 0.277·13-s + 1.45·17-s − 1.14·19-s + 0.436·21-s + 1.25·23-s − 0.192·27-s + 0.185·29-s + 1.79·31-s + 0.696·33-s + 0.493·37-s − 0.160·39-s − 1.09·41-s + 1.21·43-s + 1.02·47-s − 3/7·49-s − 0.840·51-s − 0.412·53-s + 0.662·57-s + 0.781·59-s − 0.256·61-s − 0.251·63-s + 1.09·67-s − 0.722·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 124800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.149545193\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.149545193\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - T + p T^{2} \) | 1.29.ab |
| 31 | \( 1 - 10 T + p T^{2} \) | 1.31.ak |
| 37 | \( 1 - 3 T + p T^{2} \) | 1.37.ad |
| 41 | \( 1 + 7 T + p T^{2} \) | 1.41.h |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 7 T + p T^{2} \) | 1.47.ah |
| 53 | \( 1 + 3 T + p T^{2} \) | 1.53.d |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 9 T + p T^{2} \) | 1.67.aj |
| 71 | \( 1 - 9 T + p T^{2} \) | 1.71.aj |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - T + p T^{2} \) | 1.79.ab |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
| 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
−13.38889106113787, −12.97245399809621, −12.55215626370079, −12.24098215792734, −11.59398942033024, −11.03379183269302, −10.56895186523656, −10.23580529897413, −9.743275115600401, −9.269107939951664, −8.512760546091960, −8.112961598675404, −7.623275205570935, −6.982126865028757, −6.487382075037879, −6.047154016441752, −5.504919571200535, −4.939039582646182, −4.544592122100445, −3.680432036840664, −3.216142765965387, −2.607007512283164, −2.002960717622845, −0.8493955187463551, −0.6332456251382197,
0.6332456251382197, 0.8493955187463551, 2.002960717622845, 2.607007512283164, 3.216142765965387, 3.680432036840664, 4.544592122100445, 4.939039582646182, 5.504919571200535, 6.047154016441752, 6.487382075037879, 6.982126865028757, 7.623275205570935, 8.112961598675404, 8.512760546091960, 9.269107939951664, 9.743275115600401, 10.23580529897413, 10.56895186523656, 11.03379183269302, 11.59398942033024, 12.24098215792734, 12.55215626370079, 12.97245399809621, 13.38889106113787
