| L(s) = 1 | + 4·7-s − 3·11-s − 13-s − 17-s + 19-s + 9·23-s + 6·29-s + 2·31-s − 4·37-s + 3·41-s − 7·43-s − 6·47-s + 9·49-s + 6·53-s + 6·59-s − 8·61-s − 4·67-s − 12·71-s − 2·73-s − 12·77-s − 10·79-s + 6·83-s − 4·91-s + 16·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 1.51·7-s − 0.904·11-s − 0.277·13-s − 0.242·17-s + 0.229·19-s + 1.87·23-s + 1.11·29-s + 0.359·31-s − 0.657·37-s + 0.468·41-s − 1.06·43-s − 0.875·47-s + 9/7·49-s + 0.824·53-s + 0.781·59-s − 1.02·61-s − 0.488·67-s − 1.42·71-s − 0.234·73-s − 1.36·77-s − 1.12·79-s + 0.658·83-s − 0.419·91-s + 1.62·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 244800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 244800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.852067771\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.852067771\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 17 | \( 1 + T \) | |
| good | 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 - 9 T + p T^{2} \) | 1.23.aj |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 + 7 T + p T^{2} \) | 1.43.h |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 16 T + p T^{2} \) | 1.97.aq |
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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
−13.00917385798317, −12.32252610613688, −11.77280676028541, −11.63394911296592, −10.93231643768286, −10.60893445998968, −10.28926936318179, −9.613647556649323, −9.046080151003131, −8.496532781902026, −8.318060338882991, −7.667177276159279, −7.286033623441125, −6.823723684869667, −6.201068937954375, −5.464286471462628, −5.092009105759736, −4.762813886645748, −4.330085697350718, −3.506477650343970, −2.799965917426702, −2.558678308645722, −1.639835403588358, −1.286686715144517, −0.4698551091836772,
0.4698551091836772, 1.286686715144517, 1.639835403588358, 2.558678308645722, 2.799965917426702, 3.506477650343970, 4.330085697350718, 4.762813886645748, 5.092009105759736, 5.464286471462628, 6.201068937954375, 6.823723684869667, 7.286033623441125, 7.667177276159279, 8.318060338882991, 8.496532781902026, 9.046080151003131, 9.613647556649323, 10.28926936318179, 10.60893445998968, 10.93231643768286, 11.63394911296592, 11.77280676028541, 12.32252610613688, 13.00917385798317
