| L(s) = 1 | + 3-s − 2·5-s + 9-s + 2·13-s − 2·15-s − 17-s − 25-s + 27-s − 2·29-s − 8·31-s + 6·37-s + 2·39-s + 6·41-s − 4·43-s − 2·45-s − 51-s − 14·53-s − 8·59-s + 14·61-s − 4·65-s − 4·67-s + 8·71-s − 10·73-s − 75-s + 8·79-s + 81-s − 16·83-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s + 1/3·9-s + 0.554·13-s − 0.516·15-s − 0.242·17-s − 1/5·25-s + 0.192·27-s − 0.371·29-s − 1.43·31-s + 0.986·37-s + 0.320·39-s + 0.937·41-s − 0.609·43-s − 0.298·45-s − 0.140·51-s − 1.92·53-s − 1.04·59-s + 1.79·61-s − 0.496·65-s − 0.488·67-s + 0.949·71-s − 1.17·73-s − 0.115·75-s + 0.900·79-s + 1/9·81-s − 1.75·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 159936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 159936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 17 | \( 1 + T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 14 T + p T^{2} \) | 1.53.o |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 16 T + p T^{2} \) | 1.83.q |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 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
−13.44075303984278, −13.05929143946068, −12.60279646208894, −12.18652256496166, −11.49128308283948, −11.09438899965437, −10.92506476515876, −10.08925100635532, −9.587474135508539, −9.215061476484033, −8.635036868540493, −8.144389167149080, −7.820919359403418, −7.243302437500224, −6.883706155904671, −6.088251564146884, −5.740989247776146, −4.966299602031423, −4.330220451641988, −4.027746135358983, −3.341354146472803, −3.036149937110192, −2.125862319242535, −1.660528975530520, −0.7862371393984659, 0,
0.7862371393984659, 1.660528975530520, 2.125862319242535, 3.036149937110192, 3.341354146472803, 4.027746135358983, 4.330220451641988, 4.966299602031423, 5.740989247776146, 6.088251564146884, 6.883706155904671, 7.243302437500224, 7.820919359403418, 8.144389167149080, 8.635036868540493, 9.215061476484033, 9.587474135508539, 10.08925100635532, 10.92506476515876, 11.09438899965437, 11.49128308283948, 12.18652256496166, 12.60279646208894, 13.05929143946068, 13.44075303984278
