| L(s) = 1 | + 2-s + 2·3-s + 4-s + 2·6-s − 2·7-s + 8-s + 9-s + 2·12-s − 2·14-s + 16-s + 17-s + 18-s − 4·19-s − 4·21-s − 4·23-s + 2·24-s − 4·27-s − 2·28-s − 8·29-s + 10·31-s + 32-s + 34-s + 36-s − 2·37-s − 4·38-s + 8·41-s − 4·42-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.816·6-s − 0.755·7-s + 0.353·8-s + 1/3·9-s + 0.577·12-s − 0.534·14-s + 1/4·16-s + 0.242·17-s + 0.235·18-s − 0.917·19-s − 0.872·21-s − 0.834·23-s + 0.408·24-s − 0.769·27-s − 0.377·28-s − 1.48·29-s + 1.79·31-s + 0.176·32-s + 0.171·34-s + 1/6·36-s − 0.328·37-s − 0.648·38-s + 1.24·41-s − 0.617·42-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 143650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 13 | \( 1 \) | |
| 17 | \( 1 - T \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 8 T + p T^{2} \) | 1.29.i |
| 31 | \( 1 - 10 T + p T^{2} \) | 1.31.ak |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 8 T + p T^{2} \) | 1.41.ai |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 10 T + p T^{2} \) | 1.71.ak |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 + 12 T + p T^{2} \) | 1.97.m |
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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
−13.63200699359362, −13.24994361976582, −12.80536982817098, −12.31809745191458, −11.91731198854803, −11.23290015182664, −10.80474699379187, −10.17826051616894, −9.740630365480855, −9.271466585760721, −8.796609323467923, −8.174266666278934, −7.839556181803002, −7.306396420415237, −6.633363133089187, −6.252915345956476, −5.660519322447821, −5.201931546537479, −4.201219925750669, −4.023551538720346, −3.521464126255691, −2.743009583127247, −2.489385846234733, −1.914069459167306, −0.9850008064491806, 0,
0.9850008064491806, 1.914069459167306, 2.489385846234733, 2.743009583127247, 3.521464126255691, 4.023551538720346, 4.201219925750669, 5.201931546537479, 5.660519322447821, 6.252915345956476, 6.633363133089187, 7.306396420415237, 7.839556181803002, 8.174266666278934, 8.796609323467923, 9.271466585760721, 9.740630365480855, 10.17826051616894, 10.80474699379187, 11.23290015182664, 11.91731198854803, 12.31809745191458, 12.80536982817098, 13.24994361976582, 13.63200699359362
