| L(s) = 1 | − 3-s − 7-s + 9-s + 2·11-s + 6·13-s − 4·17-s − 6·19-s + 21-s − 8·23-s − 27-s − 6·29-s + 2·31-s − 2·33-s + 4·37-s − 6·39-s + 2·41-s − 4·43-s + 8·47-s + 49-s + 4·51-s + 6·53-s + 6·57-s − 8·59-s + 10·61-s − 63-s − 8·67-s + 8·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.377·7-s + 1/3·9-s + 0.603·11-s + 1.66·13-s − 0.970·17-s − 1.37·19-s + 0.218·21-s − 1.66·23-s − 0.192·27-s − 1.11·29-s + 0.359·31-s − 0.348·33-s + 0.657·37-s − 0.960·39-s + 0.312·41-s − 0.609·43-s + 1.16·47-s + 1/7·49-s + 0.560·51-s + 0.824·53-s + 0.794·57-s − 1.04·59-s + 1.28·61-s − 0.125·63-s − 0.977·67-s + 0.963·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| good | 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 - 8 T + p T^{2} \) | 1.83.ai |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 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
−15.19178016280306, −15.00390141658959, −13.97134709426354, −13.72795864137940, −13.10764254352306, −12.67132276778809, −12.07170697039864, −11.48480736201757, −11.02005391292346, −10.60813348938332, −10.02055962949792, −9.277726904941022, −8.874284120674952, −8.258684469969085, −7.713431872157961, −6.792206762133476, −6.354925743680902, −6.083763645533152, −5.412422549301469, −4.458326831145767, −3.926351222193520, −3.658009915904351, −2.399206287488704, −1.859341472240835, −0.9239038079759451, 0,
0.9239038079759451, 1.859341472240835, 2.399206287488704, 3.658009915904351, 3.926351222193520, 4.458326831145767, 5.412422549301469, 6.083763645533152, 6.354925743680902, 6.792206762133476, 7.713431872157961, 8.258684469969085, 8.874284120674952, 9.277726904941022, 10.02055962949792, 10.60813348938332, 11.02005391292346, 11.48480736201757, 12.07170697039864, 12.67132276778809, 13.10764254352306, 13.72795864137940, 13.97134709426354, 15.00390141658959, 15.19178016280306
