| L(s) = 1 | − 4·7-s − 11-s − 13-s + 2·17-s − 4·19-s + 4·29-s + 2·31-s − 2·37-s + 6·41-s + 6·43-s − 4·47-s + 9·49-s − 6·53-s + 4·59-s + 10·61-s + 12·67-s − 4·71-s + 6·73-s + 4·77-s + 16·79-s + 18·83-s + 10·89-s + 4·91-s − 2·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 1.51·7-s − 0.301·11-s − 0.277·13-s + 0.485·17-s − 0.917·19-s + 0.742·29-s + 0.359·31-s − 0.328·37-s + 0.937·41-s + 0.914·43-s − 0.583·47-s + 9/7·49-s − 0.824·53-s + 0.520·59-s + 1.28·61-s + 1.46·67-s − 0.474·71-s + 0.702·73-s + 0.455·77-s + 1.80·79-s + 1.97·83-s + 1.05·89-s + 0.419·91-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 128700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.540844295\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.540844295\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| 13 | \( 1 + T \) | |
| good | 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 16 T + p T^{2} \) | 1.79.aq |
| 83 | \( 1 - 18 T + p T^{2} \) | 1.83.as |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 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
−13.41385897937804, −12.85056126359544, −12.69332559191675, −12.16934594278234, −11.68237992772102, −10.94873161620149, −10.54311289394413, −10.10356682486883, −9.583152399788453, −9.255785063142746, −8.673483816671235, −7.968029374002195, −7.729611623262532, −6.776626356634039, −6.644631369868998, −6.122084123959565, −5.497636302452592, −4.948904336380481, −4.294872151176223, −3.635442835720939, −3.279960105077770, −2.471352043998841, −2.202446376124285, −1.011276528156448, −0.4321112570884074,
0.4321112570884074, 1.011276528156448, 2.202446376124285, 2.471352043998841, 3.279960105077770, 3.635442835720939, 4.294872151176223, 4.948904336380481, 5.497636302452592, 6.122084123959565, 6.644631369868998, 6.776626356634039, 7.729611623262532, 7.968029374002195, 8.673483816671235, 9.255785063142746, 9.583152399788453, 10.10356682486883, 10.54311289394413, 10.94873161620149, 11.68237992772102, 12.16934594278234, 12.69332559191675, 12.85056126359544, 13.41385897937804
