| L(s) = 1 | + 3·3-s − 2·5-s − 7-s + 6·9-s + 2·13-s − 6·15-s − 3·21-s − 2·23-s − 25-s + 9·27-s − 6·29-s + 4·31-s + 2·35-s − 37-s + 6·39-s + 9·41-s + 2·43-s − 12·45-s + 9·47-s − 6·49-s + 53-s − 8·59-s + 8·61-s − 6·63-s − 4·65-s − 8·67-s − 6·69-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 0.894·5-s − 0.377·7-s + 2·9-s + 0.554·13-s − 1.54·15-s − 0.654·21-s − 0.417·23-s − 1/5·25-s + 1.73·27-s − 1.11·29-s + 0.718·31-s + 0.338·35-s − 0.164·37-s + 0.960·39-s + 1.40·41-s + 0.304·43-s − 1.78·45-s + 1.31·47-s − 6/7·49-s + 0.137·53-s − 1.04·59-s + 1.02·61-s − 0.755·63-s − 0.496·65-s − 0.977·67-s − 0.722·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 71632 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71632 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 11 | \( 1 \) | |
| 37 | \( 1 + T \) | |
| good | 3 | \( 1 - p T + p T^{2} \) | 1.3.ad |
| 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 41 | \( 1 - 9 T + p T^{2} \) | 1.41.aj |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 - 9 T + p T^{2} \) | 1.47.aj |
| 53 | \( 1 - T + p T^{2} \) | 1.53.ab |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + 9 T + p T^{2} \) | 1.71.j |
| 73 | \( 1 - T + p T^{2} \) | 1.73.ab |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 15 T + p T^{2} \) | 1.83.p |
| 89 | \( 1 - 4 T + p T^{2} \) | 1.89.ae |
| 97 | \( 1 - 4 T + p T^{2} \) | 1.97.ae |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.42925564926692, −13.87725817041510, −13.42364409466114, −13.03199058874038, −12.47793008058441, −11.97276596725341, −11.37578262650177, −10.77417366406868, −10.25062131624918, −9.585848451964496, −9.228481989639367, −8.770935123249472, −8.133794291435524, −7.860151888098758, −7.362095475081118, −6.860808277502487, −6.084142719902764, −5.523399273983824, −4.436041587421007, −4.138395980668925, −3.656241446865115, −3.065055824320154, −2.560395743602852, −1.835068168462265, −1.081338267682880, 0,
1.081338267682880, 1.835068168462265, 2.560395743602852, 3.065055824320154, 3.656241446865115, 4.138395980668925, 4.436041587421007, 5.523399273983824, 6.084142719902764, 6.860808277502487, 7.362095475081118, 7.860151888098758, 8.133794291435524, 8.770935123249472, 9.228481989639367, 9.585848451964496, 10.25062131624918, 10.77417366406868, 11.37578262650177, 11.97276596725341, 12.47793008058441, 13.03199058874038, 13.42364409466114, 13.87725817041510, 14.42925564926692
