| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s + 7-s − 8-s + 9-s + 10-s − 3·11-s + 12-s − 3·13-s − 14-s − 15-s + 16-s − 18-s + 6·19-s − 20-s + 21-s + 3·22-s + 2·23-s − 24-s − 4·25-s + 3·26-s + 27-s + 28-s − 6·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.904·11-s + 0.288·12-s − 0.832·13-s − 0.267·14-s − 0.258·15-s + 1/4·16-s − 0.235·18-s + 1.37·19-s − 0.223·20-s + 0.218·21-s + 0.639·22-s + 0.417·23-s − 0.204·24-s − 4/5·25-s + 0.588·26-s + 0.192·27-s + 0.188·28-s − 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12138 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12138 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 3 | \( 1 - T \) | |
| 7 | \( 1 - T \) | |
| 17 | \( 1 \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 + 3 T + p T^{2} \) | 1.13.d |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 11 T + p T^{2} \) | 1.37.al |
| 41 | \( 1 + 12 T + p T^{2} \) | 1.41.m |
| 43 | \( 1 - 3 T + p T^{2} \) | 1.43.ad |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 5 T + p T^{2} \) | 1.53.af |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 - 9 T + p T^{2} \) | 1.67.aj |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 15 T + p T^{2} \) | 1.73.p |
| 79 | \( 1 - 11 T + p T^{2} \) | 1.79.al |
| 83 | \( 1 - 7 T + p T^{2} \) | 1.83.ah |
| 89 | \( 1 + 13 T + p T^{2} \) | 1.89.n |
| 97 | \( 1 + 11 T + p T^{2} \) | 1.97.l |
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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
−16.54804532044997, −16.13915469669009, −15.49677801573278, −14.94291599575459, −14.69185588912418, −13.69932485671266, −13.38037733079491, −12.57027530672249, −11.98465919985189, −11.40721590576473, −10.89069111199760, −10.08519145237171, −9.685224058760290, −9.081293619920947, −8.380446278440312, −7.743250807712128, −7.439271425567797, −6.929468288875933, −5.667366537464310, −5.326701073495526, −4.341015404473173, −3.587681188438429, −2.746706622776947, −2.168605089148612, −1.131614320527872, 0,
1.131614320527872, 2.168605089148612, 2.746706622776947, 3.587681188438429, 4.341015404473173, 5.326701073495526, 5.667366537464310, 6.929468288875933, 7.439271425567797, 7.743250807712128, 8.380446278440312, 9.081293619920947, 9.685224058760290, 10.08519145237171, 10.89069111199760, 11.40721590576473, 11.98465919985189, 12.57027530672249, 13.38037733079491, 13.69932485671266, 14.69185588912418, 14.94291599575459, 15.49677801573278, 16.13915469669009, 16.54804532044997
