| L(s) = 1 | + 11-s − 2·13-s + 4·17-s − 3·19-s + 23-s − 5·25-s + 6·29-s − 2·31-s − 2·37-s + 41-s + 8·43-s + 5·47-s − 3·53-s − 5·59-s − 13·61-s + 16·73-s + 2·79-s − 6·83-s + 6·89-s − 10·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 0.301·11-s − 0.554·13-s + 0.970·17-s − 0.688·19-s + 0.208·23-s − 25-s + 1.11·29-s − 0.359·31-s − 0.328·37-s + 0.156·41-s + 1.21·43-s + 0.729·47-s − 0.412·53-s − 0.650·59-s − 1.66·61-s + 1.87·73-s + 0.225·79-s − 0.658·83-s + 0.635·89-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 23 | \( 1 - T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 11 | \( 1 - T + p T^{2} \) | 1.11.ab |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 + 3 T + p T^{2} \) | 1.19.d |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - T + p T^{2} \) | 1.41.ab |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 5 T + p T^{2} \) | 1.47.af |
| 53 | \( 1 + 3 T + p T^{2} \) | 1.53.d |
| 59 | \( 1 + 5 T + p T^{2} \) | 1.59.f |
| 61 | \( 1 + 13 T + p T^{2} \) | 1.61.n |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 16 T + p T^{2} \) | 1.73.aq |
| 79 | \( 1 - 2 T + p T^{2} \) | 1.79.ac |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−13.69511050270140, −12.83964963709808, −12.46769215251192, −12.25308908454013, −11.60979179591856, −11.16893466567249, −10.57931170628298, −10.17486230406799, −9.753947708363090, −9.023128140028796, −8.944542931094056, −8.023891504313258, −7.742178333108787, −7.302421435336716, −6.575065934133356, −6.181717778636836, −5.668089733295074, −5.043843966383140, −4.564076580470100, −3.954402175419426, −3.457354776113591, −2.758299147308936, −2.240980336790458, −1.518503510363301, −0.8431519207860697, 0,
0.8431519207860697, 1.518503510363301, 2.240980336790458, 2.758299147308936, 3.457354776113591, 3.954402175419426, 4.564076580470100, 5.043843966383140, 5.668089733295074, 6.181717778636836, 6.575065934133356, 7.302421435336716, 7.742178333108787, 8.023891504313258, 8.944542931094056, 9.023128140028796, 9.753947708363090, 10.17486230406799, 10.57931170628298, 11.16893466567249, 11.60979179591856, 12.25308908454013, 12.46769215251192, 12.83964963709808, 13.69511050270140
