| L(s) = 1 | − 3-s − 3·7-s + 9-s + 3·11-s + 7·13-s + 6·17-s + 3·21-s − 3·23-s − 27-s + 3·29-s + 2·31-s − 3·33-s + 8·37-s − 7·39-s + 41-s + 2·43-s − 8·47-s + 2·49-s − 6·51-s + 6·53-s + 7·59-s − 11·61-s − 3·63-s + 10·67-s + 3·69-s + 12·71-s + 10·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.13·7-s + 1/3·9-s + 0.904·11-s + 1.94·13-s + 1.45·17-s + 0.654·21-s − 0.625·23-s − 0.192·27-s + 0.557·29-s + 0.359·31-s − 0.522·33-s + 1.31·37-s − 1.12·39-s + 0.156·41-s + 0.304·43-s − 1.16·47-s + 2/7·49-s − 0.840·51-s + 0.824·53-s + 0.911·59-s − 1.40·61-s − 0.377·63-s + 1.22·67-s + 0.361·69-s + 1.42·71-s + 1.17·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 339600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 339600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.238944788\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.238944788\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 283 | \( 1 + T \) | |
| good | 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 - 7 T + p T^{2} \) | 1.13.ah |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 - T + p T^{2} \) | 1.41.ab |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 7 T + p T^{2} \) | 1.59.ah |
| 61 | \( 1 + 11 T + p T^{2} \) | 1.61.l |
| 67 | \( 1 - 10 T + p T^{2} \) | 1.67.ak |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - 6 T + p T^{2} \) | 1.79.ag |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 9 T + p T^{2} \) | 1.89.aj |
| 97 | \( 1 - 7 T + p T^{2} \) | 1.97.ah |
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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
−12.58663283252649, −12.13125759805023, −11.59828762164656, −11.32402674627720, −10.81896831144162, −10.13099114711038, −10.01723846851628, −9.397133877739530, −9.074672625381575, −8.359810448194981, −8.070486737234386, −7.479229217296566, −6.784390889109713, −6.384734316826888, −6.139138555769305, −5.756280389918640, −5.122888640182039, −4.448815525796005, −3.838857610845489, −3.541462475368968, −3.145584615885354, −2.307852656039280, −1.551889021966847, −0.9020331842895798, −0.6498844249884645,
0.6498844249884645, 0.9020331842895798, 1.551889021966847, 2.307852656039280, 3.145584615885354, 3.541462475368968, 3.838857610845489, 4.448815525796005, 5.122888640182039, 5.756280389918640, 6.139138555769305, 6.384734316826888, 6.784390889109713, 7.479229217296566, 8.070486737234386, 8.359810448194981, 9.074672625381575, 9.397133877739530, 10.01723846851628, 10.13099114711038, 10.81896831144162, 11.32402674627720, 11.59828762164656, 12.13125759805023, 12.58663283252649
