| L(s) = 1 | − 2-s + 4-s − 2·7-s − 8-s − 4·11-s − 4·13-s + 2·14-s + 16-s − 2·17-s + 4·22-s + 4·23-s + 4·26-s − 2·28-s + 4·31-s − 32-s + 2·34-s − 2·37-s + 41-s + 12·43-s − 4·44-s − 4·46-s − 2·47-s − 3·49-s − 4·52-s − 4·53-s + 2·56-s + 4·59-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.755·7-s − 0.353·8-s − 1.20·11-s − 1.10·13-s + 0.534·14-s + 1/4·16-s − 0.485·17-s + 0.852·22-s + 0.834·23-s + 0.784·26-s − 0.377·28-s + 0.718·31-s − 0.176·32-s + 0.342·34-s − 0.328·37-s + 0.156·41-s + 1.82·43-s − 0.603·44-s − 0.589·46-s − 0.291·47-s − 3/7·49-s − 0.554·52-s − 0.549·53-s + 0.267·56-s + 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 41 | \( 1 - T \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 10 T + p T^{2} \) | 1.71.ak |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 14 T + p T^{2} \) | 1.79.o |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 - 18 T + p T^{2} \) | 1.97.as |
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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
−15.91282309553198, −15.69037566307338, −15.18892138668608, −14.40193865684756, −13.95858619766745, −12.96761694620595, −12.84432856943507, −12.29491371177267, −11.35462921182932, −11.09656386489006, −10.21632100275019, −9.959140425261315, −9.397528207206361, −8.705394302000914, −8.154651166362366, −7.440949377848109, −7.040957967735984, −6.389603268805273, −5.616013739127324, −5.021543609697324, −4.284885038499827, −3.254762032514055, −2.655462423439712, −2.131498705115093, −0.8384661867385436, 0,
0.8384661867385436, 2.131498705115093, 2.655462423439712, 3.254762032514055, 4.284885038499827, 5.021543609697324, 5.616013739127324, 6.389603268805273, 7.040957967735984, 7.440949377848109, 8.154651166362366, 8.705394302000914, 9.397528207206361, 9.959140425261315, 10.21632100275019, 11.09656386489006, 11.35462921182932, 12.29491371177267, 12.84432856943507, 12.96761694620595, 13.95858619766745, 14.40193865684756, 15.18892138668608, 15.69037566307338, 15.91282309553198
