| L(s) = 1 | − 3-s + 7-s + 9-s − 2·11-s − 17-s − 2·19-s − 21-s − 4·23-s − 27-s − 6·29-s − 2·31-s + 2·33-s + 6·37-s − 10·41-s + 4·43-s + 8·47-s + 49-s + 51-s + 4·53-s + 2·57-s + 2·61-s + 63-s + 4·67-s + 4·69-s + 14·71-s − 4·73-s − 2·77-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.603·11-s − 0.242·17-s − 0.458·19-s − 0.218·21-s − 0.834·23-s − 0.192·27-s − 1.11·29-s − 0.359·31-s + 0.348·33-s + 0.986·37-s − 1.56·41-s + 0.609·43-s + 1.16·47-s + 1/7·49-s + 0.140·51-s + 0.549·53-s + 0.264·57-s + 0.256·61-s + 0.125·63-s + 0.488·67-s + 0.481·69-s + 1.66·71-s − 0.468·73-s − 0.227·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 17 | \( 1 + T \) | |
| good | 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 14 T + p T^{2} \) | 1.71.ao |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 18 T + p T^{2} \) | 1.89.as |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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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
−15.27791867160141, −14.72845431966111, −14.11741505114395, −13.57805898352747, −12.98327173509296, −12.65495450080426, −11.90379808696987, −11.55355951196078, −10.92617618566979, −10.51653268224613, −9.985577408011694, −9.345800616802760, −8.752016214571193, −8.102260273876001, −7.611753759459971, −7.048945166373362, −6.365214515758702, −5.771205251581813, −5.304707381606675, −4.642371400287264, −4.031073074460800, −3.409290028442309, −2.341822720271941, −1.945889758404300, −0.8818644898280137, 0,
0.8818644898280137, 1.945889758404300, 2.341822720271941, 3.409290028442309, 4.031073074460800, 4.642371400287264, 5.304707381606675, 5.771205251581813, 6.365214515758702, 7.048945166373362, 7.611753759459971, 8.102260273876001, 8.752016214571193, 9.345800616802760, 9.985577408011694, 10.51653268224613, 10.92617618566979, 11.55355951196078, 11.90379808696987, 12.65495450080426, 12.98327173509296, 13.57805898352747, 14.11741505114395, 14.72845431966111, 15.27791867160141
