| L(s) = 1 | + 2-s − 3-s + 4-s + 2·5-s − 6-s + 8-s + 9-s + 2·10-s − 12-s − 2·15-s + 16-s − 6·17-s + 18-s + 2·20-s + 4·23-s − 24-s − 25-s − 27-s − 6·29-s − 2·30-s − 4·31-s + 32-s − 6·34-s + 36-s − 2·37-s + 2·40-s + 10·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.894·5-s − 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.632·10-s − 0.288·12-s − 0.516·15-s + 1/4·16-s − 1.45·17-s + 0.235·18-s + 0.447·20-s + 0.834·23-s − 0.204·24-s − 1/5·25-s − 0.192·27-s − 1.11·29-s − 0.365·30-s − 0.718·31-s + 0.176·32-s − 1.02·34-s + 1/6·36-s − 0.328·37-s + 0.316·40-s + 1.56·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 122694 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122694 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 3 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−13.62355335779671, −13.27596180674520, −12.87146404637624, −12.52695308782715, −11.91099952372427, −11.16674516723198, −10.99737257554123, −10.74276735260341, −9.840010716662887, −9.315673777243143, −9.267589939794968, −8.288527107423575, −7.767791688140852, −7.148679771962385, −6.503107422929584, −6.410161496635615, −5.624407810888069, −5.300688490298640, −4.743280607248901, −4.189400317945676, −3.586970750741337, −2.902504147737629, −2.046915405620275, −1.917595643835461, −0.9393053600205015, 0,
0.9393053600205015, 1.917595643835461, 2.046915405620275, 2.902504147737629, 3.586970750741337, 4.189400317945676, 4.743280607248901, 5.300688490298640, 5.624407810888069, 6.410161496635615, 6.503107422929584, 7.148679771962385, 7.767791688140852, 8.288527107423575, 9.267589939794968, 9.315673777243143, 9.840010716662887, 10.74276735260341, 10.99737257554123, 11.16674516723198, 11.91099952372427, 12.52695308782715, 12.87146404637624, 13.27596180674520, 13.62355335779671
