| L(s) = 1 | − 2-s − 4-s + 3·8-s − 11-s − 6·13-s − 16-s − 6·17-s + 4·19-s + 22-s − 8·23-s + 6·26-s + 10·29-s + 4·31-s − 5·32-s + 6·34-s − 6·37-s − 4·38-s − 10·41-s − 4·43-s + 44-s + 8·46-s + 4·47-s + 6·52-s + 6·53-s − 10·58-s + 6·61-s − 4·62-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 1.06·8-s − 0.301·11-s − 1.66·13-s − 1/4·16-s − 1.45·17-s + 0.917·19-s + 0.213·22-s − 1.66·23-s + 1.17·26-s + 1.85·29-s + 0.718·31-s − 0.883·32-s + 1.02·34-s − 0.986·37-s − 0.648·38-s − 1.56·41-s − 0.609·43-s + 0.150·44-s + 1.17·46-s + 0.583·47-s + 0.832·52-s + 0.824·53-s − 1.31·58-s + 0.768·61-s − 0.508·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 2 | \( 1 + T + p T^{2} \) | 1.2.b |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−13.86606037418653, −13.40128992857603, −12.87764197412591, −12.17656176234829, −11.90478661061404, −11.49751743486696, −10.54660265216391, −10.18769103307795, −10.06192430105098, −9.433277094845558, −8.876307879607145, −8.398404318502396, −8.040138820025404, −7.451787962226454, −6.897249441499206, −6.564079915855956, −5.647845706328547, −5.097436922915343, −4.678940207401108, −4.251922937510458, −3.511466105850293, −2.699312347920845, −2.200565749662117, −1.541855565951345, −0.5893585800583939, 0,
0.5893585800583939, 1.541855565951345, 2.200565749662117, 2.699312347920845, 3.511466105850293, 4.251922937510458, 4.678940207401108, 5.097436922915343, 5.647845706328547, 6.564079915855956, 6.897249441499206, 7.451787962226454, 8.040138820025404, 8.398404318502396, 8.876307879607145, 9.433277094845558, 10.06192430105098, 10.18769103307795, 10.54660265216391, 11.49751743486696, 11.90478661061404, 12.17656176234829, 12.87764197412591, 13.40128992857603, 13.86606037418653
