| L(s) = 1 | + 5-s − 2·13-s − 2·17-s + 8·19-s + 4·23-s + 25-s − 6·29-s + 2·37-s − 6·41-s + 4·43-s − 12·47-s − 7·49-s + 6·53-s + 12·59-s − 14·61-s − 2·65-s + 12·67-s − 2·73-s − 8·79-s + 4·83-s − 2·85-s − 2·89-s + 8·95-s − 14·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.554·13-s − 0.485·17-s + 1.83·19-s + 0.834·23-s + 1/5·25-s − 1.11·29-s + 0.328·37-s − 0.937·41-s + 0.609·43-s − 1.75·47-s − 49-s + 0.824·53-s + 1.56·59-s − 1.79·61-s − 0.248·65-s + 1.46·67-s − 0.234·73-s − 0.900·79-s + 0.439·83-s − 0.216·85-s − 0.211·89-s + 0.820·95-s − 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 174240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 174240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.214075126\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.214075126\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−13.32005691089401, −12.74622968392792, −12.30980050097132, −11.65888050990962, −11.36660033633941, −10.92459943512418, −10.26645700692467, −9.741375982526167, −9.473650009081061, −9.068026372595026, −8.336753487761255, −7.934023749413206, −7.308183220090842, −6.906691917312347, −6.499164485270929, −5.655384579351261, −5.362972455382673, −4.923228225448355, −4.273883918211775, −3.581227655967067, −3.038275113622561, −2.571015882611497, −1.763870156754951, −1.292128867160636, −0.4313945754370202,
0.4313945754370202, 1.292128867160636, 1.763870156754951, 2.571015882611497, 3.038275113622561, 3.581227655967067, 4.273883918211775, 4.923228225448355, 5.362972455382673, 5.655384579351261, 6.499164485270929, 6.906691917312347, 7.308183220090842, 7.934023749413206, 8.336753487761255, 9.068026372595026, 9.473650009081061, 9.741375982526167, 10.26645700692467, 10.92459943512418, 11.36660033633941, 11.65888050990962, 12.30980050097132, 12.74622968392792, 13.32005691089401
