| L(s) = 1 | − 3-s + 5-s + 4·7-s + 9-s − 11-s + 4·13-s − 15-s − 2·17-s + 6·19-s − 4·21-s + 4·23-s + 25-s − 27-s + 4·29-s − 4·31-s + 33-s + 4·35-s − 6·37-s − 4·39-s + 12·43-s + 45-s + 9·49-s + 2·51-s − 6·53-s − 55-s − 6·57-s + 12·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1.51·7-s + 1/3·9-s − 0.301·11-s + 1.10·13-s − 0.258·15-s − 0.485·17-s + 1.37·19-s − 0.872·21-s + 0.834·23-s + 1/5·25-s − 0.192·27-s + 0.742·29-s − 0.718·31-s + 0.174·33-s + 0.676·35-s − 0.986·37-s − 0.640·39-s + 1.82·43-s + 0.149·45-s + 9/7·49-s + 0.280·51-s − 0.824·53-s − 0.134·55-s − 0.794·57-s + 1.56·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.473553169\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.473553169\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 11 | \( 1 + T \) | |
| good | 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 16 T + p T^{2} \) | 1.71.q |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 - 14 T + p T^{2} \) | 1.79.ao |
| 83 | \( 1 + 10 T + p T^{2} \) | 1.83.k |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−8.238562574490840645772847568831, −7.40832956807765292147499821738, −6.83091401334757886868118356057, −5.79951263334342655106094874110, −5.35821128057580141199890011841, −4.70363069316148912964170750480, −3.85636859070490441436807623096, −2.73962813265761704262029124929, −1.64808578833364967389975934074, −0.974160384957176716284969361156,
0.974160384957176716284969361156, 1.64808578833364967389975934074, 2.73962813265761704262029124929, 3.85636859070490441436807623096, 4.70363069316148912964170750480, 5.35821128057580141199890011841, 5.79951263334342655106094874110, 6.83091401334757886868118356057, 7.40832956807765292147499821738, 8.238562574490840645772847568831
