| L(s) = 1 | + 2-s + 3-s + 4-s + 2·5-s + 6-s + 8-s + 9-s + 2·10-s − 11-s + 12-s + 2·15-s + 16-s − 2·17-s + 18-s + 4·19-s + 2·20-s − 22-s − 8·23-s + 24-s − 25-s + 27-s − 29-s + 2·30-s − 4·31-s + 32-s − 33-s − 2·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.894·5-s + 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.632·10-s − 0.301·11-s + 0.288·12-s + 0.516·15-s + 1/4·16-s − 0.485·17-s + 0.235·18-s + 0.917·19-s + 0.447·20-s − 0.213·22-s − 1.66·23-s + 0.204·24-s − 1/5·25-s + 0.192·27-s − 0.185·29-s + 0.365·30-s − 0.718·31-s + 0.176·32-s − 0.174·33-s − 0.342·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 323466 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 323466 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 11 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| 29 | \( 1 + T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 16 T + p T^{2} \) | 1.71.aq |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 8 T + p T^{2} \) | 1.83.ai |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−12.96208426445556, −12.46409325165679, −12.07312459327525, −11.55606224894733, −11.05068217515732, −10.54145313990348, −10.06386831011632, −9.719546051203529, −9.228464524662524, −8.815428881640069, −8.109925063203785, −7.717489142433105, −7.356920182287585, −6.647201890396732, −6.268680975701791, −5.743900555282216, −5.333208439204529, −4.876980868857160, −4.129677657940828, −3.793656383983707, −3.249277627728656, −2.522849811930406, −2.177480404187950, −1.730468086653961, −0.9917433764325134, 0,
0.9917433764325134, 1.730468086653961, 2.177480404187950, 2.522849811930406, 3.249277627728656, 3.793656383983707, 4.129677657940828, 4.876980868857160, 5.333208439204529, 5.743900555282216, 6.268680975701791, 6.647201890396732, 7.356920182287585, 7.717489142433105, 8.109925063203785, 8.815428881640069, 9.228464524662524, 9.719546051203529, 10.06386831011632, 10.54145313990348, 11.05068217515732, 11.55606224894733, 12.07312459327525, 12.46409325165679, 12.96208426445556
