| L(s) = 1 | + 2·7-s − 2·11-s − 13-s + 3·17-s − 4·19-s + 6·23-s − 6·29-s + 3·31-s + 37-s − 4·43-s − 5·47-s − 3·49-s + 7·53-s + 5·59-s + 2·61-s + 5·67-s − 13·71-s + 73-s − 4·77-s + 3·79-s + 6·83-s − 2·89-s − 2·91-s + 10·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 0.755·7-s − 0.603·11-s − 0.277·13-s + 0.727·17-s − 0.917·19-s + 1.25·23-s − 1.11·29-s + 0.538·31-s + 0.164·37-s − 0.609·43-s − 0.729·47-s − 3/7·49-s + 0.961·53-s + 0.650·59-s + 0.256·61-s + 0.610·67-s − 1.54·71-s + 0.117·73-s − 0.455·77-s + 0.337·79-s + 0.658·83-s − 0.211·89-s − 0.209·91-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 266400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 266400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.910541728\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.910541728\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 37 | \( 1 - T \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 3 T + p T^{2} \) | 1.31.ad |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 5 T + p T^{2} \) | 1.47.f |
| 53 | \( 1 - 7 T + p T^{2} \) | 1.53.ah |
| 59 | \( 1 - 5 T + p T^{2} \) | 1.59.af |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 5 T + p T^{2} \) | 1.67.af |
| 71 | \( 1 + 13 T + p T^{2} \) | 1.71.n |
| 73 | \( 1 - T + p T^{2} \) | 1.73.ab |
| 79 | \( 1 - 3 T + p T^{2} \) | 1.79.ad |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−12.82922404314039, −12.37117110209707, −11.75830035135815, −11.47991127836567, −10.89312948339439, −10.60147059175430, −10.01792112202688, −9.653991789137545, −8.982519010865730, −8.592119446809857, −8.127858789240706, −7.612698413096306, −7.314985727301844, −6.614665533243843, −6.222652884885153, −5.489949002452573, −5.039705886310012, −4.862002568671505, −3.982444412450778, −3.663129752071999, −2.794158384210311, −2.497904311733223, −1.703726417868251, −1.230149285713889, −0.3781442844981474,
0.3781442844981474, 1.230149285713889, 1.703726417868251, 2.497904311733223, 2.794158384210311, 3.663129752071999, 3.982444412450778, 4.862002568671505, 5.039705886310012, 5.489949002452573, 6.222652884885153, 6.614665533243843, 7.314985727301844, 7.612698413096306, 8.127858789240706, 8.592119446809857, 8.982519010865730, 9.653991789137545, 10.01792112202688, 10.60147059175430, 10.89312948339439, 11.47991127836567, 11.75830035135815, 12.37117110209707, 12.82922404314039
