| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 7-s − 8-s + 9-s − 12-s + 13-s + 14-s + 16-s + 2·17-s − 18-s − 4·19-s + 21-s − 4·23-s + 24-s − 5·25-s − 26-s − 27-s − 28-s + 10·29-s + 2·31-s − 32-s − 2·34-s + 36-s + 8·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.288·12-s + 0.277·13-s + 0.267·14-s + 1/4·16-s + 0.485·17-s − 0.235·18-s − 0.917·19-s + 0.218·21-s − 0.834·23-s + 0.204·24-s − 25-s − 0.196·26-s − 0.192·27-s − 0.188·28-s + 1.85·29-s + 0.359·31-s − 0.176·32-s − 0.342·34-s + 1/6·36-s + 1.31·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 66066 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 66066 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−14.48881492926551, −13.97103552047473, −13.38896388138846, −12.84462772270512, −12.20791707603471, −12.05703860132792, −11.23691726702358, −11.00756573592493, −10.23915180411900, −9.955290944368863, −9.531357281838475, −8.808170157941690, −8.225489460910975, −7.868871189236362, −7.254069914758283, −6.484953629747495, −6.188502657750104, −5.823966435261240, −4.890897136691111, −4.363812768570230, −3.728910802138176, −2.934941004520474, −2.323202826519019, −1.524833864243290, −0.7978887691945390, 0,
0.7978887691945390, 1.524833864243290, 2.323202826519019, 2.934941004520474, 3.728910802138176, 4.363812768570230, 4.890897136691111, 5.823966435261240, 6.188502657750104, 6.484953629747495, 7.254069914758283, 7.868871189236362, 8.225489460910975, 8.808170157941690, 9.531357281838475, 9.955290944368863, 10.23915180411900, 11.00756573592493, 11.23691726702358, 12.05703860132792, 12.20791707603471, 12.84462772270512, 13.38896388138846, 13.97103552047473, 14.48881492926551
