| L(s) = 1 | + 3-s + 5-s − 2·7-s − 2·9-s + 11-s + 13-s + 15-s + 3·17-s − 7·19-s − 2·21-s − 6·23-s + 25-s − 5·27-s + 3·29-s + 4·31-s + 33-s − 2·35-s − 2·37-s + 39-s + 43-s − 2·45-s + 12·47-s − 3·49-s + 3·51-s − 12·53-s + 55-s − 7·57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.755·7-s − 2/3·9-s + 0.301·11-s + 0.277·13-s + 0.258·15-s + 0.727·17-s − 1.60·19-s − 0.436·21-s − 1.25·23-s + 1/5·25-s − 0.962·27-s + 0.557·29-s + 0.718·31-s + 0.174·33-s − 0.338·35-s − 0.328·37-s + 0.160·39-s + 0.152·43-s − 0.298·45-s + 1.75·47-s − 3/7·49-s + 0.420·51-s − 1.64·53-s + 0.134·55-s − 0.927·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 151360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 151360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 11 | \( 1 - T \) | |
| 43 | \( 1 - T \) | |
| good | 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 + 7 T + p T^{2} \) | 1.19.h |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 - 15 T + p T^{2} \) | 1.83.ap |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
| 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
−13.67547690255776, −13.11942712613938, −12.67169250335006, −12.18247479102875, −11.80730009907663, −11.13458182105752, −10.51566007901166, −10.30430552426629, −9.584080258736085, −9.308512821381840, −8.760712801326042, −8.161762251330076, −8.036660541362541, −7.207654904001166, −6.542601411231071, −6.172313756545443, −5.885419321873371, −5.146518628125885, −4.476507994384466, −3.841744108421913, −3.449266175010161, −2.767172641344256, −2.291242862720563, −1.729820695270760, −0.8018537233471919, 0,
0.8018537233471919, 1.729820695270760, 2.291242862720563, 2.767172641344256, 3.449266175010161, 3.841744108421913, 4.476507994384466, 5.146518628125885, 5.885419321873371, 6.172313756545443, 6.542601411231071, 7.207654904001166, 8.036660541362541, 8.161762251330076, 8.760712801326042, 9.308512821381840, 9.584080258736085, 10.30430552426629, 10.51566007901166, 11.13458182105752, 11.80730009907663, 12.18247479102875, 12.67169250335006, 13.11942712613938, 13.67547690255776
