| L(s) = 1 | + 3-s + 4·7-s + 9-s − 4·11-s + 13-s − 2·17-s + 4·19-s + 4·21-s + 27-s + 6·29-s − 4·33-s − 10·37-s + 39-s + 2·41-s − 4·43-s + 4·47-s + 9·49-s − 2·51-s + 6·53-s + 4·57-s + 4·59-s − 2·61-s + 4·63-s − 4·67-s + 8·71-s + 2·73-s − 16·77-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.51·7-s + 1/3·9-s − 1.20·11-s + 0.277·13-s − 0.485·17-s + 0.917·19-s + 0.872·21-s + 0.192·27-s + 1.11·29-s − 0.696·33-s − 1.64·37-s + 0.160·39-s + 0.312·41-s − 0.609·43-s + 0.583·47-s + 9/7·49-s − 0.280·51-s + 0.824·53-s + 0.529·57-s + 0.520·59-s − 0.256·61-s + 0.503·63-s − 0.488·67-s + 0.949·71-s + 0.234·73-s − 1.82·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.557757255\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.557757255\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−15.25251425883895, −14.36122779141423, −14.16509318186877, −13.50625223140735, −13.24278927529726, −12.30863010027744, −12.00466852633983, −11.23574666923711, −10.87844062956161, −10.23212800673204, −9.849751965367334, −8.820382330710134, −8.597922564143234, −8.083602228981987, −7.437511457438112, −7.131361558823362, −6.179015603996052, −5.382378278352650, −4.984334190783072, −4.451544398365686, −3.627512842514614, −2.904755934774069, −2.217430175688605, −1.600887400731496, −0.7046322021598749,
0.7046322021598749, 1.600887400731496, 2.217430175688605, 2.904755934774069, 3.627512842514614, 4.451544398365686, 4.984334190783072, 5.382378278352650, 6.179015603996052, 7.131361558823362, 7.437511457438112, 8.083602228981987, 8.597922564143234, 8.820382330710134, 9.849751965367334, 10.23212800673204, 10.87844062956161, 11.23574666923711, 12.00466852633983, 12.30863010027744, 13.24278927529726, 13.50625223140735, 14.16509318186877, 14.36122779141423, 15.25251425883895
