| L(s) = 1 | − 3-s − 4·7-s + 9-s + 3·11-s − 6·13-s + 4·17-s − 3·19-s + 4·21-s − 23-s − 27-s − 4·29-s + 2·31-s − 3·33-s + 3·37-s + 6·39-s + 6·41-s + 12·43-s + 9·49-s − 4·51-s − 9·53-s + 3·57-s − 7·59-s + 61-s − 4·63-s − 7·67-s + 69-s − 4·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.51·7-s + 1/3·9-s + 0.904·11-s − 1.66·13-s + 0.970·17-s − 0.688·19-s + 0.872·21-s − 0.208·23-s − 0.192·27-s − 0.742·29-s + 0.359·31-s − 0.522·33-s + 0.493·37-s + 0.960·39-s + 0.937·41-s + 1.82·43-s + 9/7·49-s − 0.560·51-s − 1.23·53-s + 0.397·57-s − 0.911·59-s + 0.128·61-s − 0.503·63-s − 0.855·67-s + 0.120·69-s − 0.468·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 398400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 398400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4000169622\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4000169622\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 83 | \( 1 + T \) | |
| good | 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 + 3 T + p T^{2} \) | 1.19.d |
| 23 | \( 1 + T + p T^{2} \) | 1.23.b |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 - 3 T + p T^{2} \) | 1.37.ad |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 + 7 T + p T^{2} \) | 1.59.h |
| 61 | \( 1 - T + p T^{2} \) | 1.61.ab |
| 67 | \( 1 + 7 T + p T^{2} \) | 1.67.h |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 89 | \( 1 - 5 T + p T^{2} \) | 1.89.af |
| 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
−12.40697309371696, −12.15870002578754, −11.69599197690091, −10.98595680155878, −10.65913341759554, −10.14638067741305, −9.613427496485473, −9.353054614704198, −9.219696327360085, −8.286531252518389, −7.648523191858636, −7.448388471438079, −6.813894955847327, −6.433057205533278, −6.022322244198530, −5.583340894179643, −5.042711207913748, −4.263345183539766, −4.130535075181663, −3.422314586355747, −2.770263009853869, −2.497514761579964, −1.612286927751797, −0.9725663972447573, −0.1915748125626917,
0.1915748125626917, 0.9725663972447573, 1.612286927751797, 2.497514761579964, 2.770263009853869, 3.422314586355747, 4.130535075181663, 4.263345183539766, 5.042711207913748, 5.583340894179643, 6.022322244198530, 6.433057205533278, 6.813894955847327, 7.448388471438079, 7.648523191858636, 8.286531252518389, 9.219696327360085, 9.353054614704198, 9.613427496485473, 10.14638067741305, 10.65913341759554, 10.98595680155878, 11.69599197690091, 12.15870002578754, 12.40697309371696
