| L(s) = 1 | + 3-s + 5-s + 4·7-s + 9-s − 4·11-s − 2·13-s + 15-s − 6·17-s − 4·19-s + 4·21-s + 8·23-s + 25-s + 27-s + 6·29-s − 4·31-s − 4·33-s + 4·35-s + 2·37-s − 2·39-s − 2·41-s + 4·43-s + 45-s + 9·49-s − 6·51-s − 10·53-s − 4·55-s − 4·57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1.51·7-s + 1/3·9-s − 1.20·11-s − 0.554·13-s + 0.258·15-s − 1.45·17-s − 0.917·19-s + 0.872·21-s + 1.66·23-s + 1/5·25-s + 0.192·27-s + 1.11·29-s − 0.718·31-s − 0.696·33-s + 0.676·35-s + 0.328·37-s − 0.320·39-s − 0.312·41-s + 0.609·43-s + 0.149·45-s + 9/7·49-s − 0.840·51-s − 1.37·53-s − 0.539·55-s − 0.529·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.328386423\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.328386423\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 67 | \( 1 + T \) | |
| good | 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 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 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 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
−14.26833017403816, −13.74625835758329, −13.26311923943748, −12.83851158391535, −12.38918200505978, −11.61243487898515, −11.01168877717769, −10.70069622966503, −10.40771558415602, −9.442285656141925, −9.081949367993550, −8.584689758511675, −7.974857534280277, −7.735518515515077, −6.931441204180363, −6.556161040054305, −5.677290252117554, −5.001824472435673, −4.716595382184146, −4.288636029556158, −3.203707492720792, −2.601364990288021, −2.121345175423030, −1.574131303520337, −0.5693001463052469,
0.5693001463052469, 1.574131303520337, 2.121345175423030, 2.601364990288021, 3.203707492720792, 4.288636029556158, 4.716595382184146, 5.001824472435673, 5.677290252117554, 6.556161040054305, 6.931441204180363, 7.735518515515077, 7.974857534280277, 8.584689758511675, 9.081949367993550, 9.442285656141925, 10.40771558415602, 10.70069622966503, 11.01168877717769, 11.61243487898515, 12.38918200505978, 12.83851158391535, 13.26311923943748, 13.74625835758329, 14.26833017403816
