| L(s) = 1 | − 3·11-s − 2·13-s − 3·17-s − 19-s − 6·23-s − 5·25-s − 6·29-s − 4·31-s − 4·37-s + 9·41-s + 43-s + 6·47-s − 12·53-s − 3·59-s − 8·61-s − 5·67-s − 12·71-s − 11·73-s + 4·79-s − 12·83-s + 6·89-s − 5·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 0.904·11-s − 0.554·13-s − 0.727·17-s − 0.229·19-s − 1.25·23-s − 25-s − 1.11·29-s − 0.718·31-s − 0.657·37-s + 1.40·41-s + 0.152·43-s + 0.875·47-s − 1.64·53-s − 0.390·59-s − 1.02·61-s − 0.610·67-s − 1.42·71-s − 1.28·73-s + 0.450·79-s − 1.31·83-s + 0.635·89-s − 0.507·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 - 9 T + p T^{2} \) | 1.41.aj |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 + 5 T + p T^{2} \) | 1.67.f |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 5 T + p T^{2} \) | 1.97.f |
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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
−14.71347254021352, −14.24684340862461, −13.74534622124177, −13.18903500337214, −12.79785028972998, −12.27652817585237, −11.72815472640277, −11.21594129418135, −10.57026510144131, −10.36260659804250, −9.525907975774960, −9.254986973339885, −8.611986568383187, −7.854039062047200, −7.607572611891998, −7.115118451683147, −6.213744192740286, −5.850422526096893, −5.330691339333628, −4.507065066420584, −4.178870548718846, −3.403566955206075, −2.659431462974629, −2.096130955009768, −1.501577316699711, 0, 0,
1.501577316699711, 2.096130955009768, 2.659431462974629, 3.403566955206075, 4.178870548718846, 4.507065066420584, 5.330691339333628, 5.850422526096893, 6.213744192740286, 7.115118451683147, 7.607572611891998, 7.854039062047200, 8.611986568383187, 9.254986973339885, 9.525907975774960, 10.36260659804250, 10.57026510144131, 11.21594129418135, 11.72815472640277, 12.27652817585237, 12.79785028972998, 13.18903500337214, 13.74534622124177, 14.24684340862461, 14.71347254021352
