| L(s) = 1 | + 2·3-s + 5-s + 2·7-s + 9-s − 11-s + 2·15-s + 3·17-s + 5·19-s + 4·21-s + 2·23-s + 25-s − 4·27-s + 2·29-s − 6·31-s − 2·33-s + 2·35-s + 3·37-s + 9·41-s + 9·43-s + 45-s + 3·47-s − 3·49-s + 6·51-s − 4·53-s − 55-s + 10·57-s + 8·59-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.447·5-s + 0.755·7-s + 1/3·9-s − 0.301·11-s + 0.516·15-s + 0.727·17-s + 1.14·19-s + 0.872·21-s + 0.417·23-s + 1/5·25-s − 0.769·27-s + 0.371·29-s − 1.07·31-s − 0.348·33-s + 0.338·35-s + 0.493·37-s + 1.40·41-s + 1.37·43-s + 0.149·45-s + 0.437·47-s − 3/7·49-s + 0.840·51-s − 0.549·53-s − 0.134·55-s + 1.32·57-s + 1.04·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.740493588\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.740493588\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 - 3 T + p T^{2} \) | 1.37.ad |
| 41 | \( 1 - 9 T + p T^{2} \) | 1.41.aj |
| 43 | \( 1 - 9 T + p T^{2} \) | 1.43.aj |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 - 11 T + p T^{2} \) | 1.67.al |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 - 18 T + p T^{2} \) | 1.83.as |
| 89 | \( 1 - 8 T + p T^{2} \) | 1.89.ai |
| 97 | \( 1 - 9 T + p T^{2} \) | 1.97.aj |
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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
−14.18792794591671, −13.73954543618640, −13.10154613773998, −12.79737577333527, −12.13272580879440, −11.48211370910336, −11.11562696929570, −10.51513291554763, −9.896912543995795, −9.376268553983181, −9.099701834655725, −8.461148970540624, −7.887571267872015, −7.600854572833417, −7.112293349827037, −6.245734121781924, −5.569764476536822, −5.275102315248174, −4.538749471844630, −3.804216752429559, −3.319364444620058, −2.583209102451611, −2.251171938121280, −1.388962625546228, −0.7678809535512294,
0.7678809535512294, 1.388962625546228, 2.251171938121280, 2.583209102451611, 3.319364444620058, 3.804216752429559, 4.538749471844630, 5.275102315248174, 5.569764476536822, 6.245734121781924, 7.112293349827037, 7.600854572833417, 7.887571267872015, 8.461148970540624, 9.099701834655725, 9.376268553983181, 9.896912543995795, 10.51513291554763, 11.11562696929570, 11.48211370910336, 12.13272580879440, 12.79737577333527, 13.10154613773998, 13.73954543618640, 14.18792794591671
