| L(s) = 1 | − 4·7-s − 6·11-s − 13-s − 6·17-s − 2·19-s − 6·23-s + 6·29-s − 2·31-s − 2·37-s + 6·41-s + 2·43-s + 12·47-s + 9·49-s + 6·53-s + 6·59-s + 2·61-s − 4·67-s − 6·71-s + 10·73-s + 24·77-s + 4·79-s + 6·89-s + 4·91-s − 2·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 1.51·7-s − 1.80·11-s − 0.277·13-s − 1.45·17-s − 0.458·19-s − 1.25·23-s + 1.11·29-s − 0.359·31-s − 0.328·37-s + 0.937·41-s + 0.304·43-s + 1.75·47-s + 9/7·49-s + 0.824·53-s + 0.781·59-s + 0.256·61-s − 0.488·67-s − 0.712·71-s + 1.17·73-s + 2.73·77-s + 0.450·79-s + 0.635·89-s + 0.419·91-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 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
−15.05451327272720, −14.22223558021067, −13.67169599848707, −13.19335155124104, −13.00044606640348, −12.19149328186992, −12.11189493797527, −11.00234998660564, −10.61382332862049, −10.27795267946377, −9.669665446267767, −9.151077117585799, −8.561353640336940, −7.987447741768773, −7.406191046616001, −6.761948628162011, −6.370542725013739, −5.657029836741539, −5.249101398558105, −4.271083913236459, −4.003007462446318, −2.973278467413552, −2.546346568278101, −2.097114405507648, −0.6244316626413631, 0,
0.6244316626413631, 2.097114405507648, 2.546346568278101, 2.973278467413552, 4.003007462446318, 4.271083913236459, 5.249101398558105, 5.657029836741539, 6.370542725013739, 6.761948628162011, 7.406191046616001, 7.987447741768773, 8.561353640336940, 9.151077117585799, 9.669665446267767, 10.27795267946377, 10.61382332862049, 11.00234998660564, 12.11189493797527, 12.19149328186992, 13.00044606640348, 13.19335155124104, 13.67169599848707, 14.22223558021067, 15.05451327272720
