| L(s) = 1 | − 2·4-s − 5·11-s + 4·16-s − 5·17-s − 19-s − 5·23-s − 5·29-s − 31-s + 10·37-s − 10·43-s + 10·44-s − 7·49-s − 5·53-s + 10·59-s + 12·61-s − 8·64-s − 5·67-s + 10·68-s + 2·76-s − 4·79-s + 15·83-s − 15·89-s + 10·92-s − 5·97-s + 10·101-s + 5·103-s − 10·107-s + ⋯ |
| L(s) = 1 | − 4-s − 1.50·11-s + 16-s − 1.21·17-s − 0.229·19-s − 1.04·23-s − 0.928·29-s − 0.179·31-s + 1.64·37-s − 1.52·43-s + 1.50·44-s − 49-s − 0.686·53-s + 1.30·59-s + 1.53·61-s − 64-s − 0.610·67-s + 1.21·68-s + 0.229·76-s − 0.450·79-s + 1.64·83-s − 1.58·89-s + 1.04·92-s − 0.507·97-s + 0.995·101-s + 0.492·103-s − 0.966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5846971447\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5846971447\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | \( 1 \) | |
| 5 | \( 1 \) | |
| 31 | \( 1 + T \) | |
| good | 2 | \( 1 + p T^{2} \) | 1.2.a |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 + 5 T + p T^{2} \) | 1.17.f |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 + 5 T + p T^{2} \) | 1.23.f |
| 29 | \( 1 + 5 T + p T^{2} \) | 1.29.f |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 5 T + p T^{2} \) | 1.53.f |
| 59 | \( 1 - 10 T + p T^{2} \) | 1.59.ak |
| 61 | \( 1 - 12 T + p T^{2} \) | 1.61.am |
| 67 | \( 1 + 5 T + p T^{2} \) | 1.67.f |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + p T^{2} \) | 1.73.a |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 - 15 T + p T^{2} \) | 1.83.ap |
| 89 | \( 1 + 15 T + p T^{2} \) | 1.89.p |
| 97 | \( 1 + 5 T + p T^{2} \) | 1.97.f |
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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
−8.170393545719501567753713652965, −7.41395744385447578882794647797, −6.48756794281593571580354007420, −5.70017460468149024733356968258, −5.07474535880573760881544752916, −4.42079435904286573611721519030, −3.70848528006472173717751843921, −2.72333841453887006520886822470, −1.86404792772923216627969938554, −0.37594103532642005402513162613,
0.37594103532642005402513162613, 1.86404792772923216627969938554, 2.72333841453887006520886822470, 3.70848528006472173717751843921, 4.42079435904286573611721519030, 5.07474535880573760881544752916, 5.70017460468149024733356968258, 6.48756794281593571580354007420, 7.41395744385447578882794647797, 8.170393545719501567753713652965
