| L(s) = 1 | + 3-s + 7-s + 9-s + 2·11-s + 6·13-s − 19-s + 21-s + 4·23-s + 27-s + 6·29-s + 10·31-s + 2·33-s + 8·37-s + 6·39-s − 2·41-s + 6·47-s + 49-s − 10·53-s − 57-s − 4·59-s − 2·61-s + 63-s − 10·67-s + 4·69-s − 8·71-s + 2·73-s + 2·77-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.377·7-s + 1/3·9-s + 0.603·11-s + 1.66·13-s − 0.229·19-s + 0.218·21-s + 0.834·23-s + 0.192·27-s + 1.11·29-s + 1.79·31-s + 0.348·33-s + 1.31·37-s + 0.960·39-s − 0.312·41-s + 0.875·47-s + 1/7·49-s − 1.37·53-s − 0.132·57-s − 0.520·59-s − 0.256·61-s + 0.125·63-s − 1.22·67-s + 0.481·69-s − 0.949·71-s + 0.234·73-s + 0.227·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 79800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 79800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.203910583\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.203910583\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 19 | \( 1 + T \) | |
| good | 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 10 T + p T^{2} \) | 1.31.ak |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 10 T + p T^{2} \) | 1.67.k |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 12 T + p T^{2} \) | 1.97.am |
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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
−13.91846890885839, −13.58288989194793, −13.16178900760324, −12.57770913678729, −11.89216077340927, −11.61932715118918, −10.90031358973317, −10.59194393615463, −9.994804763369868, −9.323455890391409, −8.846589983514185, −8.548514136019160, −7.909644392956599, −7.559501159988725, −6.637351541216943, −6.326639644689022, −5.915958671878034, −4.892225106171529, −4.529348869278826, −3.956731045740800, −3.241197441448338, −2.833831042701315, −1.982632197977682, −1.207464183038149, −0.8366492761350265,
0.8366492761350265, 1.207464183038149, 1.982632197977682, 2.833831042701315, 3.241197441448338, 3.956731045740800, 4.529348869278826, 4.892225106171529, 5.915958671878034, 6.326639644689022, 6.637351541216943, 7.559501159988725, 7.909644392956599, 8.548514136019160, 8.846589983514185, 9.323455890391409, 9.994804763369868, 10.59194393615463, 10.90031358973317, 11.61932715118918, 11.89216077340927, 12.57770913678729, 13.16178900760324, 13.58288989194793, 13.91846890885839
