| L(s) = 1 | + 3-s − 7-s + 9-s − 3·11-s + 13-s + 3·17-s − 4·19-s − 21-s − 6·23-s + 27-s + 3·29-s − 31-s − 3·33-s + 2·37-s + 39-s − 10·43-s − 3·47-s − 6·49-s + 3·51-s − 3·53-s − 4·57-s + 3·59-s + 5·61-s − 63-s − 7·67-s − 6·69-s + 2·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.377·7-s + 1/3·9-s − 0.904·11-s + 0.277·13-s + 0.727·17-s − 0.917·19-s − 0.218·21-s − 1.25·23-s + 0.192·27-s + 0.557·29-s − 0.179·31-s − 0.522·33-s + 0.328·37-s + 0.160·39-s − 1.52·43-s − 0.437·47-s − 6/7·49-s + 0.420·51-s − 0.412·53-s − 0.529·57-s + 0.390·59-s + 0.640·61-s − 0.125·63-s − 0.855·67-s − 0.722·69-s + 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 + T + p T^{2} \) | 1.31.b |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 + 3 T + p T^{2} \) | 1.53.d |
| 59 | \( 1 - 3 T + p T^{2} \) | 1.59.ad |
| 61 | \( 1 - 5 T + p T^{2} \) | 1.61.af |
| 67 | \( 1 + 7 T + p T^{2} \) | 1.67.h |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 + 15 T + p T^{2} \) | 1.83.p |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−8.221754042339034046515342071823, −7.52218298191206417199272461589, −6.63284285758713129469637358413, −5.95185215446340372734922504507, −5.06534986105564674565109366634, −4.17834025726867744408375121313, −3.34857687543109834152590964475, −2.56960853142190099040371679998, −1.58207013902956100563716211390, 0,
1.58207013902956100563716211390, 2.56960853142190099040371679998, 3.34857687543109834152590964475, 4.17834025726867744408375121313, 5.06534986105564674565109366634, 5.95185215446340372734922504507, 6.63284285758713129469637358413, 7.52218298191206417199272461589, 8.221754042339034046515342071823
