| L(s) = 1 | + 2-s + 3-s − 4-s − 5-s + 6-s − 4·7-s − 3·8-s + 9-s − 10-s + 4·11-s − 12-s + 2·13-s − 4·14-s − 15-s − 16-s + 8·17-s + 18-s + 2·19-s + 20-s − 4·21-s + 4·22-s + 6·23-s − 3·24-s + 25-s + 2·26-s + 27-s + 4·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.447·5-s + 0.408·6-s − 1.51·7-s − 1.06·8-s + 1/3·9-s − 0.316·10-s + 1.20·11-s − 0.288·12-s + 0.554·13-s − 1.06·14-s − 0.258·15-s − 1/4·16-s + 1.94·17-s + 0.235·18-s + 0.458·19-s + 0.223·20-s − 0.872·21-s + 0.852·22-s + 1.25·23-s − 0.612·24-s + 1/5·25-s + 0.392·26-s + 0.192·27-s + 0.755·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.848404695\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.848404695\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | \( 1 - T \) | |
| 5 | \( 1 + T \) | |
| 43 | \( 1 \) | |
| good | 2 | \( 1 - T + p T^{2} \) | 1.2.ab |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 8 T + p T^{2} \) | 1.17.ai |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 47 | \( 1 + 10 T + p T^{2} \) | 1.47.k |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.92842868694124, −14.69543037448213, −14.19656787967448, −13.69052256086864, −12.95833976364494, −12.72559543515226, −12.37475672585210, −11.56420520520680, −11.14191589337559, −10.12619967813802, −9.676211066944594, −9.179808863692639, −8.950284699008321, −8.069376213381579, −7.447298067729118, −6.808903499743419, −6.276144040000661, −5.532813367881774, −5.147886890574634, −3.904727023890703, −3.783465943899059, −3.328860991007097, −2.721984753621472, −1.379297225473892, −0.6188018255256282,
0.6188018255256282, 1.379297225473892, 2.721984753621472, 3.328860991007097, 3.783465943899059, 3.904727023890703, 5.147886890574634, 5.532813367881774, 6.276144040000661, 6.808903499743419, 7.447298067729118, 8.069376213381579, 8.950284699008321, 9.179808863692639, 9.676211066944594, 10.12619967813802, 11.14191589337559, 11.56420520520680, 12.37475672585210, 12.72559543515226, 12.95833976364494, 13.69052256086864, 14.19656787967448, 14.69543037448213, 14.92842868694124
