| L(s) = 1 | + 5-s − 7-s + 11-s − 17-s − 3·19-s − 6·23-s + 25-s − 9·29-s − 35-s − 11·37-s + 3·41-s + 6·43-s + 7·47-s − 6·49-s + 9·53-s + 55-s − 14·61-s + 10·67-s − 9·73-s − 77-s − 12·79-s + 12·83-s − 85-s − 3·95-s − 2·97-s − 12·101-s − 20·103-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.377·7-s + 0.301·11-s − 0.242·17-s − 0.688·19-s − 1.25·23-s + 1/5·25-s − 1.67·29-s − 0.169·35-s − 1.80·37-s + 0.468·41-s + 0.914·43-s + 1.02·47-s − 6/7·49-s + 1.23·53-s + 0.134·55-s − 1.79·61-s + 1.22·67-s − 1.05·73-s − 0.113·77-s − 1.35·79-s + 1.31·83-s − 0.108·85-s − 0.307·95-s − 0.203·97-s − 1.19·101-s − 1.97·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3060 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3060 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 17 | \( 1 + T \) | |
| good | 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 - T + p T^{2} \) | 1.11.ab |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 19 | \( 1 + 3 T + p T^{2} \) | 1.19.d |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 9 T + p T^{2} \) | 1.29.j |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 11 T + p T^{2} \) | 1.37.l |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 - 7 T + p T^{2} \) | 1.47.ah |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 - 10 T + p T^{2} \) | 1.67.ak |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 9 T + p T^{2} \) | 1.73.j |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−8.436747013224243973060129977159, −7.52171020707464993122242596659, −6.80775206465008486677025510783, −6.00586400290455266400029347819, −5.44813782080822773207079654486, −4.29805329061079409516906706806, −3.64122741659346746001000983253, −2.47600312940741915973151478205, −1.62205257101609375007641715773, 0,
1.62205257101609375007641715773, 2.47600312940741915973151478205, 3.64122741659346746001000983253, 4.29805329061079409516906706806, 5.44813782080822773207079654486, 6.00586400290455266400029347819, 6.80775206465008486677025510783, 7.52171020707464993122242596659, 8.436747013224243973060129977159
