| L(s) = 1 | − 3-s − 5-s + 3·7-s − 2·9-s − 2·11-s + 15-s − 3·17-s − 3·21-s + 6·23-s − 4·25-s + 5·27-s − 6·29-s + 2·33-s − 3·35-s + 3·37-s − 10·41-s + 9·43-s + 2·45-s + 7·47-s + 2·49-s + 3·51-s − 6·53-s + 2·55-s − 10·59-s + 10·61-s − 6·63-s + 12·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1.13·7-s − 2/3·9-s − 0.603·11-s + 0.258·15-s − 0.727·17-s − 0.654·21-s + 1.25·23-s − 4/5·25-s + 0.962·27-s − 1.11·29-s + 0.348·33-s − 0.507·35-s + 0.493·37-s − 1.56·41-s + 1.37·43-s + 0.298·45-s + 1.02·47-s + 2/7·49-s + 0.420·51-s − 0.824·53-s + 0.269·55-s − 1.30·59-s + 1.28·61-s − 0.755·63-s + 1.46·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43264 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43264 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 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 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 3 T + p T^{2} \) | 1.37.ad |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 - 9 T + p T^{2} \) | 1.43.aj |
| 47 | \( 1 - 7 T + p T^{2} \) | 1.47.ah |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 - 5 T + p T^{2} \) | 1.71.af |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 + 4 T + p T^{2} \) | 1.89.e |
| 97 | \( 1 + 18 T + p T^{2} \) | 1.97.s |
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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
−15.03271702398384, −14.47689076605128, −13.93460440796514, −13.42883334283615, −12.81843013002250, −12.28733661714819, −11.66383886810395, −11.30089717168228, −10.87624086991818, −10.61900397639165, −9.619294410794238, −9.149869871889229, −8.462783068588833, −8.071227339425309, −7.545545278609976, −6.929656261552071, −6.298295391532311, −5.482724339640274, −5.270650500271350, −4.597231371813233, −4.015902248677826, −3.205961111611607, −2.461917251822180, −1.798893462855415, −0.8326154004104429, 0,
0.8326154004104429, 1.798893462855415, 2.461917251822180, 3.205961111611607, 4.015902248677826, 4.597231371813233, 5.270650500271350, 5.482724339640274, 6.298295391532311, 6.929656261552071, 7.545545278609976, 8.071227339425309, 8.462783068588833, 9.149869871889229, 9.619294410794238, 10.61900397639165, 10.87624086991818, 11.30089717168228, 11.66383886810395, 12.28733661714819, 12.81843013002250, 13.42883334283615, 13.93460440796514, 14.47689076605128, 15.03271702398384
