| L(s) = 1 | − 3-s − 2·7-s + 9-s − 4·11-s + 2·13-s + 2·17-s − 8·19-s + 2·21-s − 4·23-s − 27-s − 6·31-s + 4·33-s − 2·37-s − 2·39-s + 6·41-s − 4·47-s − 3·49-s − 2·51-s + 8·57-s + 4·59-s − 14·61-s − 2·63-s + 4·67-s + 4·69-s + 12·71-s + 10·73-s + 8·77-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.755·7-s + 1/3·9-s − 1.20·11-s + 0.554·13-s + 0.485·17-s − 1.83·19-s + 0.436·21-s − 0.834·23-s − 0.192·27-s − 1.07·31-s + 0.696·33-s − 0.328·37-s − 0.320·39-s + 0.937·41-s − 0.583·47-s − 3/7·49-s − 0.280·51-s + 1.05·57-s + 0.520·59-s − 1.79·61-s − 0.251·63-s + 0.488·67-s + 0.481·69-s + 1.42·71-s + 1.17·73-s + 0.911·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6192847067\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6192847067\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−7.73815604164369169821342098956, −6.85101840841362537104427690689, −6.27313190421731068099902685814, −5.73364318390344893732026132208, −5.03707169329726589736408769710, −4.17603012762576770211668628573, −3.51969663080255673561059359971, −2.57264458953716636539081361760, −1.74725810428037864412471774362, −0.37131919747375929898043950705,
0.37131919747375929898043950705, 1.74725810428037864412471774362, 2.57264458953716636539081361760, 3.51969663080255673561059359971, 4.17603012762576770211668628573, 5.03707169329726589736408769710, 5.73364318390344893732026132208, 6.27313190421731068099902685814, 6.85101840841362537104427690689, 7.73815604164369169821342098956
