| L(s) = 1 | − 3-s + 5-s + 7-s − 2·9-s − 4·13-s − 15-s + 4·19-s − 21-s + 25-s + 5·27-s − 6·29-s + 10·31-s + 35-s + 8·37-s + 4·39-s − 3·41-s + 43-s − 2·45-s − 9·47-s − 6·49-s − 12·53-s − 4·57-s − 6·59-s + 11·61-s − 2·63-s − 4·65-s + 67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.377·7-s − 2/3·9-s − 1.10·13-s − 0.258·15-s + 0.917·19-s − 0.218·21-s + 1/5·25-s + 0.962·27-s − 1.11·29-s + 1.79·31-s + 0.169·35-s + 1.31·37-s + 0.640·39-s − 0.468·41-s + 0.152·43-s − 0.298·45-s − 1.31·47-s − 6/7·49-s − 1.64·53-s − 0.529·57-s − 0.781·59-s + 1.40·61-s − 0.251·63-s − 0.496·65-s + 0.122·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.440834113\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.440834113\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 10 T + p T^{2} \) | 1.31.ak |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 + 9 T + p T^{2} \) | 1.47.j |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - 11 T + p T^{2} \) | 1.61.al |
| 67 | \( 1 - T + p T^{2} \) | 1.67.ab |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 - 8 T + p T^{2} \) | 1.73.ai |
| 79 | \( 1 + 14 T + p T^{2} \) | 1.79.o |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 15 T + p T^{2} \) | 1.89.p |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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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
−7.83447357976953692984639124949, −6.83571958784720958330452612666, −6.31355124922561312930457875899, −5.55592131042436524780294585787, −5.01565923801797399813993446111, −4.48091330494385562860921794565, −3.25917783198588981349909522407, −2.63655156291518783859380784497, −1.68555557282223351044636765815, −0.58708337366800374120866693385,
0.58708337366800374120866693385, 1.68555557282223351044636765815, 2.63655156291518783859380784497, 3.25917783198588981349909522407, 4.48091330494385562860921794565, 5.01565923801797399813993446111, 5.55592131042436524780294585787, 6.31355124922561312930457875899, 6.83571958784720958330452612666, 7.83447357976953692984639124949
