| L(s) = 1 | − 5-s + 4·11-s + 2·17-s + 4·19-s − 4·23-s + 25-s + 6·29-s + 4·31-s − 6·37-s − 2·41-s + 12·43-s + 8·47-s − 7·49-s − 14·53-s − 4·55-s + 12·59-s − 2·61-s + 8·71-s + 2·73-s + 8·79-s + 12·83-s − 2·85-s + 6·89-s − 4·95-s + 10·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.20·11-s + 0.485·17-s + 0.917·19-s − 0.834·23-s + 1/5·25-s + 1.11·29-s + 0.718·31-s − 0.986·37-s − 0.312·41-s + 1.82·43-s + 1.16·47-s − 49-s − 1.92·53-s − 0.539·55-s + 1.56·59-s − 0.256·61-s + 0.949·71-s + 0.234·73-s + 0.900·79-s + 1.31·83-s − 0.216·85-s + 0.635·89-s − 0.410·95-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.801330601\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.801330601\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 14 T + p T^{2} \) | 1.53.o |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−14.22776809394838, −13.98284495587127, −13.40404660108205, −12.56280777151816, −12.15046585624139, −11.95256260101707, −11.30265587877378, −10.80353318476936, −10.16891260361176, −9.648880477285926, −9.218633144947354, −8.591381357527661, −8.065610953144658, −7.565302020933791, −6.993281442035529, −6.348624099940461, −6.020370685803276, −5.122452377424278, −4.710050861735185, −3.894765329109837, −3.580046762927951, −2.847689976966201, −2.054144553812640, −1.201256238159983, −0.6459778893542917,
0.6459778893542917, 1.201256238159983, 2.054144553812640, 2.847689976966201, 3.580046762927951, 3.894765329109837, 4.710050861735185, 5.122452377424278, 6.020370685803276, 6.348624099940461, 6.993281442035529, 7.565302020933791, 8.065610953144658, 8.591381357527661, 9.218633144947354, 9.648880477285926, 10.16891260361176, 10.80353318476936, 11.30265587877378, 11.95256260101707, 12.15046585624139, 12.56280777151816, 13.40404660108205, 13.98284495587127, 14.22776809394838
