| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 8-s + 9-s − 11-s − 12-s + 13-s + 16-s + 6·17-s − 18-s − 5·19-s + 22-s − 3·23-s + 24-s − 26-s − 27-s + 3·29-s − 5·31-s − 32-s + 33-s − 6·34-s + 36-s + 8·37-s + 5·38-s − 39-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.301·11-s − 0.288·12-s + 0.277·13-s + 1/4·16-s + 1.45·17-s − 0.235·18-s − 1.14·19-s + 0.213·22-s − 0.625·23-s + 0.204·24-s − 0.196·26-s − 0.192·27-s + 0.557·29-s − 0.898·31-s − 0.176·32-s + 0.174·33-s − 1.02·34-s + 1/6·36-s + 1.31·37-s + 0.811·38-s − 0.160·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 3 T + p T^{2} \) | 1.71.d |
| 73 | \( 1 + 8 T + p T^{2} \) | 1.73.i |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 + 9 T + p T^{2} \) | 1.83.j |
| 89 | \( 1 - 15 T + p T^{2} \) | 1.89.ap |
| 97 | \( 1 + 5 T + p T^{2} \) | 1.97.f |
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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
−14.40912098061485, −13.73971974920626, −12.96112581524310, −12.77292807273997, −12.15866192631043, −11.63694794542725, −11.25590552234095, −10.64209723804538, −10.10026592825660, −10.02086864543345, −9.156379516566835, −8.722374259700696, −8.110695966197021, −7.629124298645857, −7.229321718148565, −6.463273419181797, −5.964756705705083, −5.683963622136207, −4.846641688873303, −4.300567697382282, −3.581219916868817, −2.951900460874988, −2.182421586168379, −1.518294273941979, −0.8040975620109108, 0,
0.8040975620109108, 1.518294273941979, 2.182421586168379, 2.951900460874988, 3.581219916868817, 4.300567697382282, 4.846641688873303, 5.683963622136207, 5.964756705705083, 6.463273419181797, 7.229321718148565, 7.629124298645857, 8.110695966197021, 8.722374259700696, 9.156379516566835, 10.02086864543345, 10.10026592825660, 10.64209723804538, 11.25590552234095, 11.63694794542725, 12.15866192631043, 12.77292807273997, 12.96112581524310, 13.73971974920626, 14.40912098061485
