| L(s) = 1 | − 4·7-s + 3·11-s + 3·13-s − 17-s − 19-s + 3·23-s − 10·29-s + 6·31-s − 4·37-s − 5·41-s − 43-s − 2·47-s + 9·49-s + 14·53-s − 6·59-s − 8·61-s − 12·67-s − 12·71-s − 2·73-s − 12·77-s − 14·79-s − 6·83-s − 16·89-s − 12·91-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 1.51·7-s + 0.904·11-s + 0.832·13-s − 0.242·17-s − 0.229·19-s + 0.625·23-s − 1.85·29-s + 1.07·31-s − 0.657·37-s − 0.780·41-s − 0.152·43-s − 0.291·47-s + 9/7·49-s + 1.92·53-s − 0.781·59-s − 1.02·61-s − 1.46·67-s − 1.42·71-s − 0.234·73-s − 1.36·77-s − 1.57·79-s − 0.658·83-s − 1.69·89-s − 1.25·91-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 244800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 244800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5825537560\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5825537560\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 17 | \( 1 + T \) | |
| good | 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 - 3 T + p T^{2} \) | 1.13.ad |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 + 5 T + p T^{2} \) | 1.41.f |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 - 14 T + p T^{2} \) | 1.53.ao |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 14 T + p T^{2} \) | 1.79.o |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + 16 T + p T^{2} \) | 1.89.q |
| 97 | \( 1 + p T^{2} \) | 1.97.a |
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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
−12.98408283216360, −12.46950173279329, −11.85371208810910, −11.64955027479362, −10.97729718049013, −10.53313495832781, −10.04407757573705, −9.626384659335607, −9.054867151151444, −8.824558559189588, −8.373192100464108, −7.525787062797084, −7.068387162817016, −6.751964869827333, −6.132516239011365, −5.888855086454120, −5.308057406448451, −4.455755684300645, −4.044783674802422, −3.549229567112708, −3.049917453928617, −2.585237598781396, −1.594720227420503, −1.270385705776668, −0.2108538234987313,
0.2108538234987313, 1.270385705776668, 1.594720227420503, 2.585237598781396, 3.049917453928617, 3.549229567112708, 4.044783674802422, 4.455755684300645, 5.308057406448451, 5.888855086454120, 6.132516239011365, 6.751964869827333, 7.068387162817016, 7.525787062797084, 8.373192100464108, 8.824558559189588, 9.054867151151444, 9.626384659335607, 10.04407757573705, 10.53313495832781, 10.97729718049013, 11.64955027479362, 11.85371208810910, 12.46950173279329, 12.98408283216360
