| L(s) = 1 | − 3-s − 5-s − 7-s + 9-s − 6·11-s + 13-s + 15-s − 8·19-s + 21-s + 25-s − 27-s − 6·29-s − 2·31-s + 6·33-s + 35-s + 2·37-s − 39-s + 10·43-s − 45-s + 12·47-s + 49-s + 12·53-s + 6·55-s + 8·57-s + 2·61-s − 63-s − 65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s − 1.80·11-s + 0.277·13-s + 0.258·15-s − 1.83·19-s + 0.218·21-s + 1/5·25-s − 0.192·27-s − 1.11·29-s − 0.359·31-s + 1.04·33-s + 0.169·35-s + 0.328·37-s − 0.160·39-s + 1.52·43-s − 0.149·45-s + 1.75·47-s + 1/7·49-s + 1.64·53-s + 0.809·55-s + 1.05·57-s + 0.256·61-s − 0.125·63-s − 0.124·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 7 | \( 1 + T \) | |
| 13 | \( 1 - T \) | |
| good | 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 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
−15.60992665352685, −15.51707602892423, −14.92956151324333, −14.21441911061028, −13.42692198151871, −12.94846918800503, −12.69849548188870, −12.09775014865995, −11.26426157770444, −10.88416523794091, −10.41844757743941, −9.995018650272638, −8.961146903272682, −8.700045244324011, −7.699643751377631, −7.551305275863495, −6.739411153367130, −5.983955290356832, −5.597028538895364, −4.874888279596417, −4.149797492417215, −3.639677976810332, −2.538377169449505, −2.168701456565035, −0.7616022636976250, 0,
0.7616022636976250, 2.168701456565035, 2.538377169449505, 3.639677976810332, 4.149797492417215, 4.874888279596417, 5.597028538895364, 5.983955290356832, 6.739411153367130, 7.551305275863495, 7.699643751377631, 8.700045244324011, 8.961146903272682, 9.995018650272638, 10.41844757743941, 10.88416523794091, 11.26426157770444, 12.09775014865995, 12.69849548188870, 12.94846918800503, 13.42692198151871, 14.21441911061028, 14.92956151324333, 15.51707602892423, 15.60992665352685
