| L(s) = 1 | + 3-s + 9-s − 5·11-s − 2·13-s − 17-s + 4·19-s + 7·23-s + 27-s + 6·29-s + 5·31-s − 5·33-s − 2·39-s + 10·41-s − 12·43-s + 7·47-s − 51-s + 4·53-s + 4·57-s + 9·59-s + 3·61-s − 11·67-s + 7·69-s + 7·71-s − 11·73-s − 10·79-s + 81-s + 5·83-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s − 1.50·11-s − 0.554·13-s − 0.242·17-s + 0.917·19-s + 1.45·23-s + 0.192·27-s + 1.11·29-s + 0.898·31-s − 0.870·33-s − 0.320·39-s + 1.56·41-s − 1.82·43-s + 1.02·47-s − 0.140·51-s + 0.549·53-s + 0.529·57-s + 1.17·59-s + 0.384·61-s − 1.34·67-s + 0.842·69-s + 0.830·71-s − 1.28·73-s − 1.12·79-s + 1/9·81-s + 0.548·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 249900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 249900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| 17 | \( 1 + T \) | |
| good | 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 7 T + p T^{2} \) | 1.23.ah |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 + p T^{2} \) | 1.37.a |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 - 7 T + p T^{2} \) | 1.47.ah |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 - 9 T + p T^{2} \) | 1.59.aj |
| 61 | \( 1 - 3 T + p T^{2} \) | 1.61.ad |
| 67 | \( 1 + 11 T + p T^{2} \) | 1.67.l |
| 71 | \( 1 - 7 T + p T^{2} \) | 1.71.ah |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 - 5 T + p T^{2} \) | 1.83.af |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−13.19590738452567, −12.76037414795242, −12.07605589381600, −11.88523676849333, −11.14488097243302, −10.66875067270346, −10.35371311673650, −9.680327211861100, −9.573471524941537, −8.734027825985932, −8.438399583726222, −7.971597498973539, −7.434528703809707, −7.029911978206057, −6.656544673216869, −5.743050027618317, −5.426505693160672, −4.834146675430724, −4.491891500356552, −3.789974237242598, −2.985049492090682, −2.742939966525233, −2.401218277252952, −1.405078236324249, −0.8689420130392929, 0,
0.8689420130392929, 1.405078236324249, 2.401218277252952, 2.742939966525233, 2.985049492090682, 3.789974237242598, 4.491891500356552, 4.834146675430724, 5.426505693160672, 5.743050027618317, 6.656544673216869, 7.029911978206057, 7.434528703809707, 7.971597498973539, 8.438399583726222, 8.734027825985932, 9.573471524941537, 9.680327211861100, 10.35371311673650, 10.66875067270346, 11.14488097243302, 11.88523676849333, 12.07605589381600, 12.76037414795242, 13.19590738452567
