| L(s) = 1 | − 2-s − 4-s + 5-s + 5·7-s + 3·8-s − 10-s + 4·13-s − 5·14-s − 16-s + 6·17-s + 2·19-s − 20-s + 3·23-s − 4·25-s − 4·26-s − 5·28-s + 4·29-s − 7·31-s − 5·32-s − 6·34-s + 5·35-s + 5·37-s − 2·38-s + 3·40-s − 3·41-s − 7·43-s − 3·46-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.447·5-s + 1.88·7-s + 1.06·8-s − 0.316·10-s + 1.10·13-s − 1.33·14-s − 1/4·16-s + 1.45·17-s + 0.458·19-s − 0.223·20-s + 0.625·23-s − 4/5·25-s − 0.784·26-s − 0.944·28-s + 0.742·29-s − 1.25·31-s − 0.883·32-s − 1.02·34-s + 0.845·35-s + 0.821·37-s − 0.324·38-s + 0.474·40-s − 0.468·41-s − 1.06·43-s − 0.442·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72963 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72963 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.766497458\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.766497458\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 11 | \( 1 \) | |
| 67 | \( 1 - T \) | |
| good | 2 | \( 1 + T + p T^{2} \) | 1.2.b |
| 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 7 | \( 1 - 5 T + p T^{2} \) | 1.7.af |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 + 7 T + p T^{2} \) | 1.31.h |
| 37 | \( 1 - 5 T + p T^{2} \) | 1.37.af |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 + 7 T + p T^{2} \) | 1.43.h |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 5 T + p T^{2} \) | 1.53.af |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 13 T + p T^{2} \) | 1.73.an |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - T + p T^{2} \) | 1.83.ab |
| 89 | \( 1 + 4 T + p T^{2} \) | 1.89.e |
| 97 | \( 1 + 12 T + p T^{2} \) | 1.97.m |
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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
−13.97505761693259, −13.83364354330713, −13.20527798962982, −12.63584110905483, −11.94721763689433, −11.43913355365577, −11.03084336436442, −10.56470411794888, −10.00034077081168, −9.510552861662855, −9.011976432738572, −8.342633826738719, −8.077336811496671, −7.746237834404025, −7.057679500788599, −6.320930782766749, −5.437938966437345, −5.276581043305222, −4.762648141343215, −3.890905668851896, −3.572129616747844, −2.493288587336453, −1.576924580256097, −1.405254544540228, −0.6939268521130325,
0.6939268521130325, 1.405254544540228, 1.576924580256097, 2.493288587336453, 3.572129616747844, 3.890905668851896, 4.762648141343215, 5.276581043305222, 5.437938966437345, 6.320930782766749, 7.057679500788599, 7.746237834404025, 8.077336811496671, 8.342633826738719, 9.011976432738572, 9.510552861662855, 10.00034077081168, 10.56470411794888, 11.03084336436442, 11.43913355365577, 11.94721763689433, 12.63584110905483, 13.20527798962982, 13.83364354330713, 13.97505761693259
