| L(s) = 1 | + 2·7-s + 11-s + 4·17-s + 4·19-s + 4·23-s − 6·29-s − 6·37-s − 10·41-s − 2·43-s − 4·47-s − 3·49-s − 2·53-s − 12·59-s − 6·61-s − 8·67-s + 8·73-s + 2·77-s − 8·79-s − 18·83-s − 14·89-s − 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + 8·119-s + ⋯ |
| L(s) = 1 | + 0.755·7-s + 0.301·11-s + 0.970·17-s + 0.917·19-s + 0.834·23-s − 1.11·29-s − 0.986·37-s − 1.56·41-s − 0.304·43-s − 0.583·47-s − 3/7·49-s − 0.274·53-s − 1.56·59-s − 0.768·61-s − 0.977·67-s + 0.936·73-s + 0.227·77-s − 0.900·79-s − 1.97·83-s − 1.48·89-s − 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + 0.733·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 8 T + p T^{2} \) | 1.73.ai |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 18 T + p T^{2} \) | 1.83.s |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 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
−15.93677686634400, −15.25587589127747, −14.91057160007980, −14.32244214070138, −13.77867848466300, −13.38848796714763, −12.43381133136818, −12.24545226287075, −11.31864121979682, −11.25221468641312, −10.38718221482921, −9.785724387477451, −9.309766071312229, −8.549874984502166, −8.107821044003551, −7.354609187469196, −7.018458092036247, −6.115906246576388, −5.408262005342831, −5.017268681710795, −4.256921791560652, −3.366133925328983, −2.970388142823574, −1.653078628469374, −1.376035658749064, 0,
1.376035658749064, 1.653078628469374, 2.970388142823574, 3.366133925328983, 4.256921791560652, 5.017268681710795, 5.408262005342831, 6.115906246576388, 7.018458092036247, 7.354609187469196, 8.107821044003551, 8.549874984502166, 9.309766071312229, 9.785724387477451, 10.38718221482921, 11.25221468641312, 11.31864121979682, 12.24545226287075, 12.43381133136818, 13.38848796714763, 13.77867848466300, 14.32244214070138, 14.91057160007980, 15.25587589127747, 15.93677686634400
