| L(s) = 1 | + 2·3-s + 9-s − 11-s + 13-s + 4·17-s + 19-s + 7·23-s − 4·27-s − 9·29-s + 31-s − 2·33-s + 8·37-s + 2·39-s + 8·41-s − 11·43-s + 8·47-s − 7·49-s + 8·51-s − 2·53-s + 2·57-s + 4·59-s − 2·61-s + 10·67-s + 14·69-s + 7·71-s + 16·73-s + 4·79-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1/3·9-s − 0.301·11-s + 0.277·13-s + 0.970·17-s + 0.229·19-s + 1.45·23-s − 0.769·27-s − 1.67·29-s + 0.179·31-s − 0.348·33-s + 1.31·37-s + 0.320·39-s + 1.24·41-s − 1.67·43-s + 1.16·47-s − 49-s + 1.12·51-s − 0.274·53-s + 0.264·57-s + 0.520·59-s − 0.256·61-s + 1.22·67-s + 1.68·69-s + 0.830·71-s + 1.87·73-s + 0.450·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.293845642\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.293845642\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 - 7 T + p T^{2} \) | 1.23.ah |
| 29 | \( 1 + 9 T + p T^{2} \) | 1.29.j |
| 31 | \( 1 - T + p T^{2} \) | 1.31.ab |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 - 8 T + p T^{2} \) | 1.41.ai |
| 43 | \( 1 + 11 T + p T^{2} \) | 1.43.l |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 10 T + p T^{2} \) | 1.67.ak |
| 71 | \( 1 - 7 T + p T^{2} \) | 1.71.ah |
| 73 | \( 1 - 16 T + p T^{2} \) | 1.73.aq |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 - T + p T^{2} \) | 1.83.ab |
| 89 | \( 1 + 5 T + p T^{2} \) | 1.89.f |
| 97 | \( 1 - 3 T + p T^{2} \) | 1.97.ad |
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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
−7.87352502678659796327676787964, −7.31169229320086984319186369483, −6.47566522787760568537561525512, −5.59330370476647393906558558036, −5.04189551060581128620852411386, −3.96902723450432327856214115180, −3.38071041263620786598716909896, −2.74174052841916080898443935592, −1.92185399885296536570320881293, −0.841106961884761491180944789459,
0.841106961884761491180944789459, 1.92185399885296536570320881293, 2.74174052841916080898443935592, 3.38071041263620786598716909896, 3.96902723450432327856214115180, 5.04189551060581128620852411386, 5.59330370476647393906558558036, 6.47566522787760568537561525512, 7.31169229320086984319186369483, 7.87352502678659796327676787964
