| L(s) = 1 | + 2·5-s + 2·7-s − 3·13-s − 17-s + 5·19-s − 25-s + 6·29-s + 4·35-s − 6·37-s + 4·41-s + 11·43-s − 9·47-s − 3·49-s + 6·53-s + 4·59-s + 4·61-s − 6·65-s − 9·67-s − 8·71-s − 16·73-s + 10·79-s − 9·83-s − 2·85-s − 3·89-s − 6·91-s + 10·95-s − 10·97-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 0.755·7-s − 0.832·13-s − 0.242·17-s + 1.14·19-s − 1/5·25-s + 1.11·29-s + 0.676·35-s − 0.986·37-s + 0.624·41-s + 1.67·43-s − 1.31·47-s − 3/7·49-s + 0.824·53-s + 0.520·59-s + 0.512·61-s − 0.744·65-s − 1.09·67-s − 0.949·71-s − 1.87·73-s + 1.12·79-s − 0.987·83-s − 0.216·85-s − 0.317·89-s − 0.628·91-s + 1.02·95-s − 1.01·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 296208 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 296208 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 11 | \( 1 \) | |
| 17 | \( 1 + T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 13 | \( 1 + 3 T + p T^{2} \) | 1.13.d |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 4 T + p T^{2} \) | 1.41.ae |
| 43 | \( 1 - 11 T + p T^{2} \) | 1.43.al |
| 47 | \( 1 + 9 T + p T^{2} \) | 1.47.j |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 + 9 T + p T^{2} \) | 1.67.j |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 16 T + p T^{2} \) | 1.73.q |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 + 9 T + p T^{2} \) | 1.83.j |
| 89 | \( 1 + 3 T + p T^{2} \) | 1.89.d |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−12.96705886717096, −12.42746443658625, −11.90161002911340, −11.72249939886857, −11.03329734065813, −10.64491898066612, −10.08847185746190, −9.728650659266431, −9.389053408744377, −8.723109302005076, −8.400699554890086, −7.767487954084508, −7.273325408411292, −7.008452313583359, −6.202344898727272, −5.865639849036434, −5.284354137294390, −4.947928898218719, −4.396737852031592, −3.869865279452817, −2.969742738874631, −2.707284653470023, −2.000141593334246, −1.505927352729755, −0.9264953367367685, 0,
0.9264953367367685, 1.505927352729755, 2.000141593334246, 2.707284653470023, 2.969742738874631, 3.869865279452817, 4.396737852031592, 4.947928898218719, 5.284354137294390, 5.865639849036434, 6.202344898727272, 7.008452313583359, 7.273325408411292, 7.767487954084508, 8.400699554890086, 8.723109302005076, 9.389053408744377, 9.728650659266431, 10.08847185746190, 10.64491898066612, 11.03329734065813, 11.72249939886857, 11.90161002911340, 12.42746443658625, 12.96705886717096
