| L(s) = 1 | + 3-s − 5-s + 9-s − 11-s − 6·13-s − 15-s + 3·17-s − 6·19-s + 23-s − 4·25-s + 27-s − 10·29-s + 8·31-s − 33-s + 2·37-s − 6·39-s + 5·41-s − 6·43-s − 45-s + 3·47-s + 3·51-s + 6·53-s + 55-s − 6·57-s − 8·59-s + 9·61-s + 6·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1/3·9-s − 0.301·11-s − 1.66·13-s − 0.258·15-s + 0.727·17-s − 1.37·19-s + 0.208·23-s − 4/5·25-s + 0.192·27-s − 1.85·29-s + 1.43·31-s − 0.174·33-s + 0.328·37-s − 0.960·39-s + 0.780·41-s − 0.914·43-s − 0.149·45-s + 0.437·47-s + 0.420·51-s + 0.824·53-s + 0.134·55-s − 0.794·57-s − 1.04·59-s + 1.15·61-s + 0.744·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 103488 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 103488 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 - T + p T^{2} \) | 1.23.ab |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 5 T + p T^{2} \) | 1.41.af |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 - 9 T + p T^{2} \) | 1.61.aj |
| 67 | \( 1 + 13 T + p T^{2} \) | 1.67.n |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 - 5 T + p T^{2} \) | 1.79.af |
| 83 | \( 1 - 15 T + p T^{2} \) | 1.83.ap |
| 89 | \( 1 + 18 T + p T^{2} \) | 1.89.s |
| 97 | \( 1 + 9 T + p T^{2} \) | 1.97.j |
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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
−14.00511332921808, −13.47463420405267, −12.97151570615580, −12.44354691734868, −12.19627364651563, −11.50798438695259, −11.11885839984584, −10.32481204424541, −10.07428573817630, −9.511124578954799, −9.086810498255013, −8.360159608112471, −7.989990782872648, −7.482416688913157, −7.162412611572943, −6.453210667076906, −5.797063092034524, −5.267389849491082, −4.519297679891508, −4.253488029867660, −3.496649738113550, −2.939873905033059, −2.222187302497787, −1.916125558933228, −0.7575027142912297, 0,
0.7575027142912297, 1.916125558933228, 2.222187302497787, 2.939873905033059, 3.496649738113550, 4.253488029867660, 4.519297679891508, 5.267389849491082, 5.797063092034524, 6.453210667076906, 7.162412611572943, 7.482416688913157, 7.989990782872648, 8.360159608112471, 9.086810498255013, 9.511124578954799, 10.07428573817630, 10.32481204424541, 11.11885839984584, 11.50798438695259, 12.19627364651563, 12.44354691734868, 12.97151570615580, 13.47463420405267, 14.00511332921808
