| L(s) = 1 | − 2·5-s + 7-s − 4·11-s − 2·13-s − 4·19-s − 8·23-s − 25-s + 6·29-s − 2·35-s + 2·37-s + 10·41-s + 4·43-s + 49-s − 6·53-s + 8·55-s − 4·59-s − 6·61-s + 4·65-s + 12·67-s + 8·71-s + 6·73-s − 4·77-s − 12·83-s + 6·89-s − 2·91-s + 8·95-s − 2·97-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.377·7-s − 1.20·11-s − 0.554·13-s − 0.917·19-s − 1.66·23-s − 1/5·25-s + 1.11·29-s − 0.338·35-s + 0.328·37-s + 1.56·41-s + 0.609·43-s + 1/7·49-s − 0.824·53-s + 1.07·55-s − 0.520·59-s − 0.768·61-s + 0.496·65-s + 1.46·67-s + 0.949·71-s + 0.702·73-s − 0.455·77-s − 1.31·83-s + 0.635·89-s − 0.209·91-s + 0.820·95-s − 0.203·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 291312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 291312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7030188373\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7030188373\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 17 | \( 1 \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−12.50600035400855, −12.43728232671535, −11.82938487505323, −11.20163259846990, −11.04409953498601, −10.45417035863923, −9.927628821469026, −9.701637535881233, −8.845654527674099, −8.460530438564796, −7.963133921898200, −7.627009497620560, −7.427954317546072, −6.496656573869284, −6.170200181507369, −5.619276090640942, −4.934922202138430, −4.591370249493031, −4.081670678659481, −3.632467815623032, −2.843806843127369, −2.360572375815981, −1.957855917290653, −0.9617108115997332, −0.2543364531731940,
0.2543364531731940, 0.9617108115997332, 1.957855917290653, 2.360572375815981, 2.843806843127369, 3.632467815623032, 4.081670678659481, 4.591370249493031, 4.934922202138430, 5.619276090640942, 6.170200181507369, 6.496656573869284, 7.427954317546072, 7.627009497620560, 7.963133921898200, 8.460530438564796, 8.845654527674099, 9.701637535881233, 9.927628821469026, 10.45417035863923, 11.04409953498601, 11.20163259846990, 11.82938487505323, 12.43728232671535, 12.50600035400855
