| L(s) = 1 | − 7-s + 11-s − 2·13-s − 6·17-s + 4·19-s + 8·23-s − 2·29-s − 4·31-s − 2·37-s − 10·41-s + 4·43-s − 8·47-s + 49-s + 10·53-s + 4·59-s − 10·61-s + 8·71-s + 6·73-s − 77-s + 4·79-s + 8·83-s + 6·89-s + 2·91-s − 14·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 0.377·7-s + 0.301·11-s − 0.554·13-s − 1.45·17-s + 0.917·19-s + 1.66·23-s − 0.371·29-s − 0.718·31-s − 0.328·37-s − 1.56·41-s + 0.609·43-s − 1.16·47-s + 1/7·49-s + 1.37·53-s + 0.520·59-s − 1.28·61-s + 0.949·71-s + 0.702·73-s − 0.113·77-s + 0.450·79-s + 0.878·83-s + 0.635·89-s + 0.209·91-s − 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 138600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 138600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 11 | \( 1 - T \) | |
| good | 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.ae |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 - 8 T + p T^{2} \) | 1.83.ai |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−13.58486910569233, −13.23965623419484, −12.72822437539715, −12.27038616069103, −11.68207009020626, −11.30267379562406, −10.80625571814558, −10.35052375296810, −9.672485514631088, −9.303242923631392, −8.900485127909712, −8.426151595483190, −7.691411137731676, −7.162533830915724, −6.823407116659901, −6.377406177973220, −5.641744798968979, −4.950312852445718, −4.878160697264899, −3.905898239011201, −3.496012188769714, −2.858063029530672, −2.252123499614785, −1.599727163160233, −0.7838406484781164, 0,
0.7838406484781164, 1.599727163160233, 2.252123499614785, 2.858063029530672, 3.496012188769714, 3.905898239011201, 4.878160697264899, 4.950312852445718, 5.641744798968979, 6.377406177973220, 6.823407116659901, 7.162533830915724, 7.691411137731676, 8.426151595483190, 8.900485127909712, 9.303242923631392, 9.672485514631088, 10.35052375296810, 10.80625571814558, 11.30267379562406, 11.68207009020626, 12.27038616069103, 12.72822437539715, 13.23965623419484, 13.58486910569233
