| L(s) = 1 | + 3-s + 5-s + 4·7-s − 2·9-s + 6·11-s + 15-s − 2·19-s + 4·21-s + 23-s + 25-s − 5·27-s + 9·29-s − 5·31-s + 6·33-s + 4·35-s − 2·37-s + 9·41-s − 4·43-s − 2·45-s + 3·47-s + 9·49-s − 6·53-s + 6·55-s − 2·57-s + 2·61-s − 8·63-s + 10·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1.51·7-s − 2/3·9-s + 1.80·11-s + 0.258·15-s − 0.458·19-s + 0.872·21-s + 0.208·23-s + 1/5·25-s − 0.962·27-s + 1.67·29-s − 0.898·31-s + 1.04·33-s + 0.676·35-s − 0.328·37-s + 1.40·41-s − 0.609·43-s − 0.298·45-s + 0.437·47-s + 9/7·49-s − 0.824·53-s + 0.809·55-s − 0.264·57-s + 0.256·61-s − 1.00·63-s + 1.22·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 77740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77740 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.271812680\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.271812680\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| 23 | \( 1 - T \) | |
| good | 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 29 | \( 1 - 9 T + p T^{2} \) | 1.29.aj |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 9 T + p T^{2} \) | 1.41.aj |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 10 T + p T^{2} \) | 1.67.ak |
| 71 | \( 1 - 3 T + p T^{2} \) | 1.71.ad |
| 73 | \( 1 - 7 T + p T^{2} \) | 1.73.ah |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−14.08715249699785, −13.84425638331457, −13.16589686802214, −12.33910830269717, −12.10301436102098, −11.48441532118436, −11.00338511369433, −10.76487123841585, −9.843890010634737, −9.348742461806489, −8.961084492802744, −8.396521440365344, −8.146879992714284, −7.468746765947192, −6.693879234326508, −6.433954467076034, −5.594313712586818, −5.207285966432891, −4.403430738342874, −4.075304911278940, −3.325246627866991, −2.602600725147542, −1.961555834899865, −1.447542211816882, −0.7520591832752914,
0.7520591832752914, 1.447542211816882, 1.961555834899865, 2.602600725147542, 3.325246627866991, 4.075304911278940, 4.403430738342874, 5.207285966432891, 5.594313712586818, 6.433954467076034, 6.693879234326508, 7.468746765947192, 8.146879992714284, 8.396521440365344, 8.961084492802744, 9.348742461806489, 9.843890010634737, 10.76487123841585, 11.00338511369433, 11.48441532118436, 12.10301436102098, 12.33910830269717, 13.16589686802214, 13.84425638331457, 14.08715249699785
