| L(s) = 1 | − 3·3-s − 2·7-s + 6·9-s − 4·11-s − 13-s + 17-s − 7·19-s + 6·21-s − 6·23-s − 9·27-s + 7·29-s + 31-s + 12·33-s − 8·37-s + 3·39-s + 10·41-s + 2·43-s + 3·47-s − 3·49-s − 3·51-s − 3·53-s + 21·57-s + 5·59-s − 7·61-s − 12·63-s − 14·67-s + 18·69-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 0.755·7-s + 2·9-s − 1.20·11-s − 0.277·13-s + 0.242·17-s − 1.60·19-s + 1.30·21-s − 1.25·23-s − 1.73·27-s + 1.29·29-s + 0.179·31-s + 2.08·33-s − 1.31·37-s + 0.480·39-s + 1.56·41-s + 0.304·43-s + 0.437·47-s − 3/7·49-s − 0.420·51-s − 0.412·53-s + 2.78·57-s + 0.650·59-s − 0.896·61-s − 1.51·63-s − 1.71·67-s + 2.16·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 17 | \( 1 - T \) | |
| good | 3 | \( 1 + p T + p T^{2} \) | 1.3.d |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 19 | \( 1 + 7 T + p T^{2} \) | 1.19.h |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 7 T + p T^{2} \) | 1.29.ah |
| 31 | \( 1 - T + p T^{2} \) | 1.31.ab |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 + 3 T + p T^{2} \) | 1.53.d |
| 59 | \( 1 - 5 T + p T^{2} \) | 1.59.af |
| 61 | \( 1 + 7 T + p T^{2} \) | 1.61.h |
| 67 | \( 1 + 14 T + p T^{2} \) | 1.67.o |
| 71 | \( 1 + 3 T + p T^{2} \) | 1.71.d |
| 73 | \( 1 + 15 T + p T^{2} \) | 1.73.p |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + T + p T^{2} \) | 1.89.b |
| 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
−15.86946081622599, −15.58089161370798, −14.89761114952868, −14.10731061131357, −13.50433197739791, −12.81948367871229, −12.56937041379419, −12.09983522099973, −11.57617190321921, −10.75343472894409, −10.44629889668217, −10.20644975917670, −9.474430993032857, −8.672182468439851, −7.965545102496076, −7.368390047003422, −6.701476640756243, −6.147969656093695, −5.851449960826423, −5.127478052553156, −4.513868705169395, −4.036551201241183, −2.957493377412860, −2.250044954274459, −1.239470448185257, 0, 0,
1.239470448185257, 2.250044954274459, 2.957493377412860, 4.036551201241183, 4.513868705169395, 5.127478052553156, 5.851449960826423, 6.147969656093695, 6.701476640756243, 7.368390047003422, 7.965545102496076, 8.672182468439851, 9.474430993032857, 10.20644975917670, 10.44629889668217, 10.75343472894409, 11.57617190321921, 12.09983522099973, 12.56937041379419, 12.81948367871229, 13.50433197739791, 14.10731061131357, 14.89761114952868, 15.58089161370798, 15.86946081622599
