| L(s) = 1 | − 3-s − 2·4-s − 5-s + 3·7-s − 2·9-s − 4·11-s + 2·12-s + 2·13-s + 15-s + 4·16-s − 5·17-s − 4·19-s + 2·20-s − 3·21-s + 2·23-s − 4·25-s + 5·27-s − 6·28-s + 5·29-s + 4·33-s − 3·35-s + 4·36-s − 10·37-s − 2·39-s + 9·41-s − 43-s + 8·44-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s − 0.447·5-s + 1.13·7-s − 2/3·9-s − 1.20·11-s + 0.577·12-s + 0.554·13-s + 0.258·15-s + 16-s − 1.21·17-s − 0.917·19-s + 0.447·20-s − 0.654·21-s + 0.417·23-s − 4/5·25-s + 0.962·27-s − 1.13·28-s + 0.928·29-s + 0.696·33-s − 0.507·35-s + 2/3·36-s − 1.64·37-s − 0.320·39-s + 1.40·41-s − 0.152·43-s + 1.20·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5461 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5461 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4933324357\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4933324357\) |
| \(L(\frac{3}{2})\) |
|
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 | 43 | \( 1 + T \) | |
| 127 | \( 1 - T \) | |
| good | 2 | \( 1 + p T^{2} \) | 1.2.a |
| 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 5 T + p T^{2} \) | 1.17.f |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 - 5 T + p T^{2} \) | 1.29.af |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 - 9 T + p T^{2} \) | 1.41.aj |
| 47 | \( 1 + 13 T + p T^{2} \) | 1.47.n |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 2 T + p T^{2} \) | 1.83.ac |
| 89 | \( 1 - 7 T + p T^{2} \) | 1.89.ah |
| 97 | \( 1 + 16 T + p T^{2} \) | 1.97.q |
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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
−8.286448279596499719489306241797, −7.74230774701709757931735272446, −6.61489421138399010298982051619, −5.88854355780383798015287745018, −4.95676835726116893655046236323, −4.82052656953891632090909441236, −3.90395447586184915953974724483, −2.87963347072967921328978658308, −1.75290471158204675173427738463, −0.38495690831672330251391784244,
0.38495690831672330251391784244, 1.75290471158204675173427738463, 2.87963347072967921328978658308, 3.90395447586184915953974724483, 4.82052656953891632090909441236, 4.95676835726116893655046236323, 5.88854355780383798015287745018, 6.61489421138399010298982051619, 7.74230774701709757931735272446, 8.286448279596499719489306241797
