| L(s) = 1 | + 2·7-s − 3·11-s − 4·13-s + 6·17-s − 7·19-s − 6·23-s + 3·29-s − 5·31-s − 4·37-s − 3·41-s − 8·43-s − 3·49-s + 6·53-s + 3·59-s − 14·61-s − 2·67-s + 15·71-s + 10·73-s − 6·77-s − 8·79-s − 15·89-s − 8·91-s − 8·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 0.755·7-s − 0.904·11-s − 1.10·13-s + 1.45·17-s − 1.60·19-s − 1.25·23-s + 0.557·29-s − 0.898·31-s − 0.657·37-s − 0.468·41-s − 1.21·43-s − 3/7·49-s + 0.824·53-s + 0.390·59-s − 1.79·61-s − 0.244·67-s + 1.78·71-s + 1.17·73-s − 0.683·77-s − 0.900·79-s − 1.58·89-s − 0.838·91-s − 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 129600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6456953528\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6456953528\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 7 T + p T^{2} \) | 1.19.h |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 3 T + p T^{2} \) | 1.59.ad |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 - 15 T + p T^{2} \) | 1.71.ap |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 15 T + p T^{2} \) | 1.89.p |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−13.59952485900270, −12.88807325259870, −12.36814495627254, −12.24583430100490, −11.62479750875153, −10.99414822713559, −10.57400650096609, −10.07290813372633, −9.815877770011821, −9.109580967818572, −8.305967897921831, −8.160314024898645, −7.758168769961897, −7.016022971953503, −6.695359816847127, −5.710897242846235, −5.587013260153159, −4.817091843741530, −4.540376527107462, −3.744235245873947, −3.197136913813510, −2.405302822033926, −1.995171121491172, −1.350796787537349, −0.2321894647259165,
0.2321894647259165, 1.350796787537349, 1.995171121491172, 2.405302822033926, 3.197136913813510, 3.744235245873947, 4.540376527107462, 4.817091843741530, 5.587013260153159, 5.710897242846235, 6.695359816847127, 7.016022971953503, 7.758168769961897, 8.160314024898645, 8.305967897921831, 9.109580967818572, 9.815877770011821, 10.07290813372633, 10.57400650096609, 10.99414822713559, 11.62479750875153, 12.24583430100490, 12.36814495627254, 12.88807325259870, 13.59952485900270
