| L(s) = 1 | − 4·5-s + 2·11-s − 5·13-s + 6·17-s − 4·19-s + 6·23-s + 11·25-s − 6·29-s − 7·31-s + 7·37-s + 2·41-s + 7·43-s + 2·47-s − 6·53-s − 8·55-s − 6·59-s + 9·61-s + 20·65-s + 7·67-s − 8·71-s − 10·73-s − 79-s − 14·83-s − 24·85-s + 12·89-s + 16·95-s + 15·97-s + ⋯ |
| L(s) = 1 | − 1.78·5-s + 0.603·11-s − 1.38·13-s + 1.45·17-s − 0.917·19-s + 1.25·23-s + 11/5·25-s − 1.11·29-s − 1.25·31-s + 1.15·37-s + 0.312·41-s + 1.06·43-s + 0.291·47-s − 0.824·53-s − 1.07·55-s − 0.781·59-s + 1.15·61-s + 2.48·65-s + 0.855·67-s − 0.949·71-s − 1.17·73-s − 0.112·79-s − 1.53·83-s − 2.60·85-s + 1.27·89-s + 1.64·95-s + 1.52·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9375404757\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9375404757\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| good | 5 | \( 1 + 4 T + p T^{2} \) | 1.5.e |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 + 5 T + p T^{2} \) | 1.13.f |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 7 T + p T^{2} \) | 1.31.h |
| 37 | \( 1 - 7 T + p T^{2} \) | 1.37.ah |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 7 T + p T^{2} \) | 1.43.ah |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - 9 T + p T^{2} \) | 1.61.aj |
| 67 | \( 1 - 7 T + p T^{2} \) | 1.67.ah |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + T + p T^{2} \) | 1.79.b |
| 83 | \( 1 + 14 T + p T^{2} \) | 1.83.o |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 97 | \( 1 - 15 T + p T^{2} \) | 1.97.ap |
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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
−15.55632334977715, −14.83064923776987, −14.61578945321711, −14.44819093506366, −13.14898197991532, −12.71205195542492, −12.30874410700265, −11.79573618200660, −11.20548808936192, −10.86502015713333, −10.07364932532710, −9.318473538652612, −8.962948151570343, −8.107901370727957, −7.582678726859549, −7.330614758539957, −6.686598606785197, −5.725612382636720, −5.091930003305483, −4.344491361304970, −3.924535977017106, −3.210328129207268, −2.552959461264987, −1.358659384971096, −0.4206151719111524,
0.4206151719111524, 1.358659384971096, 2.552959461264987, 3.210328129207268, 3.924535977017106, 4.344491361304970, 5.091930003305483, 5.725612382636720, 6.686598606785197, 7.330614758539957, 7.582678726859549, 8.107901370727957, 8.962948151570343, 9.318473538652612, 10.07364932532710, 10.86502015713333, 11.20548808936192, 11.79573618200660, 12.30874410700265, 12.71205195542492, 13.14898197991532, 14.44819093506366, 14.61578945321711, 14.83064923776987, 15.55632334977715
