| L(s) = 1 | + 2·7-s − 2·11-s − 13-s − 2·17-s + 2·19-s + 8·23-s − 6·29-s + 2·31-s + 2·37-s + 2·41-s − 6·47-s − 3·49-s + 10·53-s + 14·59-s + 10·61-s − 2·67-s − 6·71-s − 2·73-s − 4·77-s + 12·79-s + 6·83-s + 18·89-s − 2·91-s + 14·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 0.755·7-s − 0.603·11-s − 0.277·13-s − 0.485·17-s + 0.458·19-s + 1.66·23-s − 1.11·29-s + 0.359·31-s + 0.328·37-s + 0.312·41-s − 0.875·47-s − 3/7·49-s + 1.37·53-s + 1.82·59-s + 1.28·61-s − 0.244·67-s − 0.712·71-s − 0.234·73-s − 0.455·77-s + 1.35·79-s + 0.658·83-s + 1.90·89-s − 0.209·91-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 187200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.093633871\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.093633871\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 - 14 T + p T^{2} \) | 1.59.ao |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 18 T + p T^{2} \) | 1.89.as |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−13.15893213303010, −12.92683147255835, −12.03237404061737, −11.67057433677928, −11.24915904875829, −10.89349403193066, −10.25323687119412, −9.942311913337007, −9.177967156622382, −8.916418855458683, −8.368051346137411, −7.744413374814563, −7.476450546787215, −6.895017720125932, −6.414170762022407, −5.663109743836321, −5.201005901199033, −4.870846929380639, −4.318400566106183, −3.558382211825186, −3.129370597196116, −2.271284441316822, −2.051046057734989, −1.059025736097215, −0.5657397400821620,
0.5657397400821620, 1.059025736097215, 2.051046057734989, 2.271284441316822, 3.129370597196116, 3.558382211825186, 4.318400566106183, 4.870846929380639, 5.201005901199033, 5.663109743836321, 6.414170762022407, 6.895017720125932, 7.476450546787215, 7.744413374814563, 8.368051346137411, 8.916418855458683, 9.177967156622382, 9.942311913337007, 10.25323687119412, 10.89349403193066, 11.24915904875829, 11.67057433677928, 12.03237404061737, 12.92683147255835, 13.15893213303010
