| L(s) = 1 | − 3·3-s − 2·7-s + 6·9-s − 11-s + 4·13-s − 5·17-s − 19-s + 6·21-s + 2·23-s − 9·27-s + 8·29-s + 10·31-s + 3·33-s − 6·37-s − 12·39-s − 3·41-s + 4·43-s − 4·47-s − 3·49-s + 15·51-s + 6·53-s + 3·57-s − 8·59-s − 10·61-s − 12·63-s − 67-s − 6·69-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 0.755·7-s + 2·9-s − 0.301·11-s + 1.10·13-s − 1.21·17-s − 0.229·19-s + 1.30·21-s + 0.417·23-s − 1.73·27-s + 1.48·29-s + 1.79·31-s + 0.522·33-s − 0.986·37-s − 1.92·39-s − 0.468·41-s + 0.609·43-s − 0.583·47-s − 3/7·49-s + 2.10·51-s + 0.824·53-s + 0.397·57-s − 1.04·59-s − 1.28·61-s − 1.51·63-s − 0.122·67-s − 0.722·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| good | 3 | \( 1 + p T + p T^{2} \) | 1.3.d |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + 5 T + p T^{2} \) | 1.17.f |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 - 10 T + p T^{2} \) | 1.31.ak |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + T + p T^{2} \) | 1.67.b |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 3 T + p T^{2} \) | 1.73.d |
| 79 | \( 1 - 6 T + p T^{2} \) | 1.79.ag |
| 83 | \( 1 + 13 T + p T^{2} \) | 1.83.n |
| 89 | \( 1 + 9 T + p T^{2} \) | 1.89.j |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−9.082991202004099760475659306955, −8.244316517577727107918756546062, −6.95268732326195632978359768612, −6.45582489283827185575337605654, −5.94043223933086726670693722273, −4.89083596978582713873600092229, −4.24835504027946759092675905890, −2.92697909945045420809700317317, −1.27038127949828397780341120625, 0,
1.27038127949828397780341120625, 2.92697909945045420809700317317, 4.24835504027946759092675905890, 4.89083596978582713873600092229, 5.94043223933086726670693722273, 6.45582489283827185575337605654, 6.95268732326195632978359768612, 8.244316517577727107918756546062, 9.082991202004099760475659306955
