| L(s) = 1 | + 2-s − 2·3-s + 4-s − 5-s − 2·6-s + 5·7-s + 8-s + 9-s − 10-s + 3·11-s − 2·12-s − 5·13-s + 5·14-s + 2·15-s + 16-s + 17-s + 18-s + 19-s − 20-s − 10·21-s + 3·22-s + 4·23-s − 2·24-s + 25-s − 5·26-s + 4·27-s + 5·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.447·5-s − 0.816·6-s + 1.88·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s + 0.904·11-s − 0.577·12-s − 1.38·13-s + 1.33·14-s + 0.516·15-s + 1/4·16-s + 0.242·17-s + 0.235·18-s + 0.229·19-s − 0.223·20-s − 2.18·21-s + 0.639·22-s + 0.834·23-s − 0.408·24-s + 1/5·25-s − 0.980·26-s + 0.769·27-s + 0.944·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 163370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 163370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.387495987\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.387495987\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 5 | \( 1 + T \) | |
| 17 | \( 1 - T \) | |
| 31 | \( 1 \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 7 | \( 1 - 5 T + p T^{2} \) | 1.7.af |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 + 5 T + p T^{2} \) | 1.13.f |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 4 T + p T^{2} \) | 1.41.e |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 3 T + p T^{2} \) | 1.53.d |
| 59 | \( 1 + 7 T + p T^{2} \) | 1.59.h |
| 61 | \( 1 + 5 T + p T^{2} \) | 1.61.f |
| 67 | \( 1 - 14 T + p T^{2} \) | 1.67.ao |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 + 11 T + p T^{2} \) | 1.79.l |
| 83 | \( 1 + T + p T^{2} \) | 1.83.b |
| 89 | \( 1 - 16 T + p T^{2} \) | 1.89.aq |
| 97 | \( 1 + 17 T + p T^{2} \) | 1.97.r |
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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
−13.15759084683723, −12.58719079842447, −12.13046202401104, −11.69128120633542, −11.61384483485271, −11.10542383673150, −10.66379416389915, −10.19363092114858, −9.444911445287634, −8.946308039789504, −8.242961989177132, −7.834523477897197, −7.356106907229090, −6.762183528977051, −6.503183573130941, −5.482757520643336, −5.320961825890138, −4.931424814751884, −4.466990957131664, −3.928401443224832, −3.213022483147899, −2.500037410159156, −1.666907342305055, −1.366546363235042, −0.4324003792280718,
0.4324003792280718, 1.366546363235042, 1.666907342305055, 2.500037410159156, 3.213022483147899, 3.928401443224832, 4.466990957131664, 4.931424814751884, 5.320961825890138, 5.482757520643336, 6.503183573130941, 6.762183528977051, 7.356106907229090, 7.834523477897197, 8.242961989177132, 8.946308039789504, 9.444911445287634, 10.19363092114858, 10.66379416389915, 11.10542383673150, 11.61384483485271, 11.69128120633542, 12.13046202401104, 12.58719079842447, 13.15759084683723
