| L(s) = 1 | − 3-s + 3·7-s + 9-s − 11-s − 5·17-s + 19-s − 3·21-s − 3·23-s − 27-s + 2·29-s + 6·31-s + 33-s − 3·37-s − 41-s + 4·43-s + 7·47-s + 2·49-s + 5·51-s − 8·53-s − 57-s − 11·59-s − 8·61-s + 3·63-s + 8·67-s + 3·69-s − 71-s − 8·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.13·7-s + 1/3·9-s − 0.301·11-s − 1.21·17-s + 0.229·19-s − 0.654·21-s − 0.625·23-s − 0.192·27-s + 0.371·29-s + 1.07·31-s + 0.174·33-s − 0.493·37-s − 0.156·41-s + 0.609·43-s + 1.02·47-s + 2/7·49-s + 0.700·51-s − 1.09·53-s − 0.132·57-s − 1.43·59-s − 1.02·61-s + 0.377·63-s + 0.977·67-s + 0.361·69-s − 0.118·71-s − 0.936·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.684848575\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.684848575\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 + 5 T + p T^{2} \) | 1.17.f |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 + T + p T^{2} \) | 1.41.b |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 7 T + p T^{2} \) | 1.47.ah |
| 53 | \( 1 + 8 T + p T^{2} \) | 1.53.i |
| 59 | \( 1 + 11 T + p T^{2} \) | 1.59.l |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + T + p T^{2} \) | 1.71.b |
| 73 | \( 1 + 8 T + p T^{2} \) | 1.73.i |
| 79 | \( 1 - 17 T + p T^{2} \) | 1.79.ar |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 97 | \( 1 - 9 T + p T^{2} \) | 1.97.aj |
| 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
−16.15904630040268, −15.63730597372259, −15.28202499203063, −14.55814770444390, −13.81360448091020, −13.65107243833302, −12.73154742769473, −12.14308279210318, −11.73101341208809, −10.94536701654597, −10.78328590347817, −10.02989779112676, −9.264452410344421, −8.664612222245534, −7.942841063507783, −7.558866095096817, −6.651104840841825, −6.160022979519961, −5.383423624194371, −4.642591857209045, −4.416405518614582, −3.318655758906318, −2.320071464398153, −1.652301461535524, −0.6057962640421855,
0.6057962640421855, 1.652301461535524, 2.320071464398153, 3.318655758906318, 4.416405518614582, 4.642591857209045, 5.383423624194371, 6.160022979519961, 6.651104840841825, 7.558866095096817, 7.942841063507783, 8.664612222245534, 9.264452410344421, 10.02989779112676, 10.78328590347817, 10.94536701654597, 11.73101341208809, 12.14308279210318, 12.73154742769473, 13.65107243833302, 13.81360448091020, 14.55814770444390, 15.28202499203063, 15.63730597372259, 16.15904630040268
