| L(s) = 1 | − 4·5-s − 7-s + 11-s + 6·13-s + 2·19-s + 4·23-s + 11·25-s + 2·29-s + 2·31-s + 4·35-s + 2·37-s + 4·43-s + 6·47-s + 49-s − 2·53-s − 4·55-s + 14·61-s − 24·65-s + 12·67-s + 8·71-s − 4·73-s − 77-s − 8·79-s + 2·83-s + 14·89-s − 6·91-s − 8·95-s + ⋯ |
| L(s) = 1 | − 1.78·5-s − 0.377·7-s + 0.301·11-s + 1.66·13-s + 0.458·19-s + 0.834·23-s + 11/5·25-s + 0.371·29-s + 0.359·31-s + 0.676·35-s + 0.328·37-s + 0.609·43-s + 0.875·47-s + 1/7·49-s − 0.274·53-s − 0.539·55-s + 1.79·61-s − 2.97·65-s + 1.46·67-s + 0.949·71-s − 0.468·73-s − 0.113·77-s − 0.900·79-s + 0.219·83-s + 1.48·89-s − 0.628·91-s − 0.820·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.788157530\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.788157530\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 11 | \( 1 - T \) | |
| good | 5 | \( 1 + 4 T + p T^{2} \) | 1.5.e |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 2 T + p T^{2} \) | 1.83.ac |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 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
−15.67575861839741, −15.11034157820158, −14.55005552949271, −13.92764942355743, −13.26356302025437, −12.78815298473628, −12.17645277377064, −11.67607869140489, −11.14398407911992, −10.86343044034429, −10.09566757903756, −9.237405571614405, −8.776711552140450, −8.200646607091586, −7.790576662671874, −6.972365614570151, −6.655110794419205, −5.834725941722024, −5.070586729034815, −4.267680201214145, −3.780570423742223, −3.346032181242030, −2.558202612736574, −1.165791796976558, −0.6537611495166747,
0.6537611495166747, 1.165791796976558, 2.558202612736574, 3.346032181242030, 3.780570423742223, 4.267680201214145, 5.070586729034815, 5.834725941722024, 6.655110794419205, 6.972365614570151, 7.790576662671874, 8.200646607091586, 8.776711552140450, 9.237405571614405, 10.09566757903756, 10.86343044034429, 11.14398407911992, 11.67607869140489, 12.17645277377064, 12.78815298473628, 13.26356302025437, 13.92764942355743, 14.55005552949271, 15.11034157820158, 15.67575861839741
