| L(s) = 1 | − 2-s + 4-s − 5-s − 8-s − 3·9-s + 10-s − 4·11-s + 16-s − 3·17-s + 3·18-s − 20-s + 4·22-s − 4·23-s − 4·25-s − 29-s + 4·31-s − 32-s + 3·34-s − 3·36-s − 3·37-s + 40-s − 9·41-s − 8·43-s − 4·44-s + 3·45-s + 4·46-s − 8·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.353·8-s − 9-s + 0.316·10-s − 1.20·11-s + 1/4·16-s − 0.727·17-s + 0.707·18-s − 0.223·20-s + 0.852·22-s − 0.834·23-s − 4/5·25-s − 0.185·29-s + 0.718·31-s − 0.176·32-s + 0.514·34-s − 1/2·36-s − 0.493·37-s + 0.158·40-s − 1.40·41-s − 1.21·43-s − 0.603·44-s + 0.447·45-s + 0.589·46-s − 1.16·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16562 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16562 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + T + p T^{2} \) | 1.29.b |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 + 9 T + p T^{2} \) | 1.41.j |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 7 T + p T^{2} \) | 1.61.h |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 11 T + p T^{2} \) | 1.73.al |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.61406423662685, −15.77569732735607, −15.54381402883572, −15.06615959919807, −14.26603550845166, −13.68025768857831, −13.24090335474356, −12.38834769913543, −11.92301691914987, −11.30003714414636, −10.96372163617308, −10.16387437296085, −9.821816527807418, −8.977646798065315, −8.361766127653985, −8.028313378962876, −7.521303760950472, −6.563435371709086, −6.194905063128782, −5.272033396468020, −4.823997288595571, −3.746343133880131, −3.092875757849823, −2.372951159438782, −1.609315477169007, 0, 0,
1.609315477169007, 2.372951159438782, 3.092875757849823, 3.746343133880131, 4.823997288595571, 5.272033396468020, 6.194905063128782, 6.563435371709086, 7.521303760950472, 8.028313378962876, 8.361766127653985, 8.977646798065315, 9.821816527807418, 10.16387437296085, 10.96372163617308, 11.30003714414636, 11.92301691914987, 12.38834769913543, 13.24090335474356, 13.68025768857831, 14.26603550845166, 15.06615959919807, 15.54381402883572, 15.77569732735607, 16.61406423662685
