| L(s) = 1 | − 5-s + 4·11-s + 6·13-s + 4·17-s − 6·19-s + 25-s − 6·29-s − 4·31-s − 8·37-s + 10·41-s − 2·43-s − 10·47-s + 14·53-s − 4·55-s − 4·59-s − 8·61-s − 6·65-s + 6·67-s − 2·71-s + 10·73-s − 16·79-s − 8·83-s − 4·85-s + 2·89-s + 6·95-s − 2·97-s + 101-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.20·11-s + 1.66·13-s + 0.970·17-s − 1.37·19-s + 1/5·25-s − 1.11·29-s − 0.718·31-s − 1.31·37-s + 1.56·41-s − 0.304·43-s − 1.45·47-s + 1.92·53-s − 0.539·55-s − 0.520·59-s − 1.02·61-s − 0.744·65-s + 0.733·67-s − 0.237·71-s + 1.17·73-s − 1.80·79-s − 0.878·83-s − 0.433·85-s + 0.211·89-s + 0.615·95-s − 0.203·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 141120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 141120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| good | 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 + 10 T + p T^{2} \) | 1.47.k |
| 53 | \( 1 - 14 T + p T^{2} \) | 1.53.ao |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 - 6 T + p T^{2} \) | 1.67.ag |
| 71 | \( 1 + 2 T + p T^{2} \) | 1.71.c |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 + 8 T + p T^{2} \) | 1.83.i |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 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
−13.66049397539754, −13.08431808929757, −12.68156585778997, −12.25375953948917, −11.65507636356482, −11.18886374158048, −10.95204684339206, −10.32815919742048, −9.806915587557590, −9.086294546189491, −8.849780660934082, −8.353004014886578, −7.871804909061594, −7.150335272576544, −6.834398986929285, −6.053993299302800, −5.923544012011809, −5.195369425887798, −4.397622045567873, −3.900889358660949, −3.627863811010313, −3.048450276130672, −2.037524614755417, −1.532520612737893, −0.9284077282030424, 0,
0.9284077282030424, 1.532520612737893, 2.037524614755417, 3.048450276130672, 3.627863811010313, 3.900889358660949, 4.397622045567873, 5.195369425887798, 5.923544012011809, 6.053993299302800, 6.834398986929285, 7.150335272576544, 7.871804909061594, 8.353004014886578, 8.849780660934082, 9.086294546189491, 9.806915587557590, 10.32815919742048, 10.95204684339206, 11.18886374158048, 11.65507636356482, 12.25375953948917, 12.68156585778997, 13.08431808929757, 13.66049397539754
