| L(s) = 1 | + 2·3-s + 5-s + 9-s − 4·11-s + 2·13-s + 2·15-s − 4·19-s − 4·23-s + 25-s − 4·27-s + 29-s − 4·31-s − 8·33-s + 8·37-s + 4·39-s − 2·41-s − 2·43-s + 45-s + 2·47-s − 7·49-s + 14·53-s − 4·55-s − 8·57-s − 4·59-s − 2·61-s + 2·65-s + 4·67-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.447·5-s + 1/3·9-s − 1.20·11-s + 0.554·13-s + 0.516·15-s − 0.917·19-s − 0.834·23-s + 1/5·25-s − 0.769·27-s + 0.185·29-s − 0.718·31-s − 1.39·33-s + 1.31·37-s + 0.640·39-s − 0.312·41-s − 0.304·43-s + 0.149·45-s + 0.291·47-s − 49-s + 1.92·53-s − 0.539·55-s − 1.05·57-s − 0.520·59-s − 0.256·61-s + 0.248·65-s + 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 29 | \( 1 - T \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 - 14 T + p T^{2} \) | 1.53.ao |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 12 T + p T^{2} \) | 1.73.m |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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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
−7.64645176412998051295674141773, −6.74767650322653115866862816627, −5.94243374697725297711657227774, −5.41841474305090309175653044136, −4.41514848466740474103794523314, −3.75503784584083282805677446262, −2.83688006836849780274397501180, −2.38260142152677273193226584705, −1.53248742561401324339137470807, 0,
1.53248742561401324339137470807, 2.38260142152677273193226584705, 2.83688006836849780274397501180, 3.75503784584083282805677446262, 4.41514848466740474103794523314, 5.41841474305090309175653044136, 5.94243374697725297711657227774, 6.74767650322653115866862816627, 7.64645176412998051295674141773
