| L(s) = 1 | + 2·3-s + 4·7-s + 9-s + 2·13-s − 6·17-s + 7·19-s + 8·21-s − 3·23-s − 5·25-s − 4·27-s − 5·31-s + 37-s + 4·39-s + 6·41-s + 43-s − 6·47-s + 9·49-s − 12·51-s + 9·53-s + 14·57-s + 12·59-s + 8·61-s + 4·63-s − 2·67-s − 6·69-s + 6·71-s + 11·73-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1.51·7-s + 1/3·9-s + 0.554·13-s − 1.45·17-s + 1.60·19-s + 1.74·21-s − 0.625·23-s − 25-s − 0.769·27-s − 0.898·31-s + 0.164·37-s + 0.640·39-s + 0.937·41-s + 0.152·43-s − 0.875·47-s + 9/7·49-s − 1.68·51-s + 1.23·53-s + 1.85·57-s + 1.56·59-s + 1.02·61-s + 0.503·63-s − 0.244·67-s − 0.722·69-s + 0.712·71-s + 1.28·73-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
| \(L(1)\) |
\(\approx\) |
\(4.913303397\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.913303397\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 7 T + p T^{2} \) | 1.19.ah |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 - 11 T + p T^{2} \) | 1.73.al |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
| 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
−14.09212001416451, −13.64352396801086, −13.44800981530143, −12.76803621334292, −11.96399732057973, −11.53543064431092, −11.17105063521881, −10.75671509311737, −9.806380957513743, −9.568730390358942, −8.815534999335315, −8.560370163414877, −7.991570993548839, −7.620927831784223, −7.128661719324699, −6.341767261887936, −5.596061147763162, −5.202786193211658, −4.484262964980469, −3.808744397980727, −3.555474226422555, −2.452708206554034, −2.191194708070121, −1.543848388755712, −0.6753726322631511,
0.6753726322631511, 1.543848388755712, 2.191194708070121, 2.452708206554034, 3.555474226422555, 3.808744397980727, 4.484262964980469, 5.202786193211658, 5.596061147763162, 6.341767261887936, 7.128661719324699, 7.620927831784223, 7.991570993548839, 8.560370163414877, 8.815534999335315, 9.568730390358942, 9.806380957513743, 10.75671509311737, 11.17105063521881, 11.53543064431092, 11.96399732057973, 12.76803621334292, 13.44800981530143, 13.64352396801086, 14.09212001416451
