| L(s) = 1 | + 2·7-s − 5·13-s + 6·17-s − 6·19-s + 23-s − 5·25-s − 9·29-s − 3·31-s − 8·37-s − 3·41-s + 8·43-s + 7·47-s − 3·49-s + 2·53-s + 4·59-s − 10·61-s − 8·67-s + 7·71-s + 9·73-s + 6·79-s − 14·83-s − 16·89-s − 10·91-s + 6·97-s − 6·101-s − 14·103-s + 14·107-s + ⋯ |
| L(s) = 1 | + 0.755·7-s − 1.38·13-s + 1.45·17-s − 1.37·19-s + 0.208·23-s − 25-s − 1.67·29-s − 0.538·31-s − 1.31·37-s − 0.468·41-s + 1.21·43-s + 1.02·47-s − 3/7·49-s + 0.274·53-s + 0.520·59-s − 1.28·61-s − 0.977·67-s + 0.830·71-s + 1.05·73-s + 0.675·79-s − 1.53·83-s − 1.69·89-s − 1.04·91-s + 0.609·97-s − 0.597·101-s − 1.37·103-s + 1.35·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 23 | \( 1 - T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 5 T + p T^{2} \) | 1.13.f |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 29 | \( 1 + 9 T + p T^{2} \) | 1.29.j |
| 31 | \( 1 + 3 T + p T^{2} \) | 1.31.d |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 7 T + p T^{2} \) | 1.47.ah |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 7 T + p T^{2} \) | 1.71.ah |
| 73 | \( 1 - 9 T + p T^{2} \) | 1.73.aj |
| 79 | \( 1 - 6 T + p T^{2} \) | 1.79.ag |
| 83 | \( 1 + 14 T + p T^{2} \) | 1.83.o |
| 89 | \( 1 + 16 T + p T^{2} \) | 1.89.q |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−8.110478443981525229338745686859, −7.57574393299778347582875795759, −6.96311085153201898297894475156, −5.77320141887774085455693828185, −5.30781951251755615786230141863, −4.38617246092236815909135983678, −3.59004246894799179150414569661, −2.40410911892081301411643930480, −1.61191613063293050436333394676, 0,
1.61191613063293050436333394676, 2.40410911892081301411643930480, 3.59004246894799179150414569661, 4.38617246092236815909135983678, 5.30781951251755615786230141863, 5.77320141887774085455693828185, 6.96311085153201898297894475156, 7.57574393299778347582875795759, 8.110478443981525229338745686859
