| L(s) = 1 | + 3-s + 2·7-s + 9-s − 4·11-s − 4·13-s + 2·17-s − 8·19-s + 2·21-s + 4·23-s + 27-s + 8·29-s − 4·31-s − 4·33-s + 2·37-s − 4·39-s − 41-s − 4·43-s − 2·47-s − 3·49-s + 2·51-s + 4·53-s − 8·57-s + 12·59-s + 6·61-s + 2·63-s − 16·67-s + 4·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.755·7-s + 1/3·9-s − 1.20·11-s − 1.10·13-s + 0.485·17-s − 1.83·19-s + 0.436·21-s + 0.834·23-s + 0.192·27-s + 1.48·29-s − 0.718·31-s − 0.696·33-s + 0.328·37-s − 0.640·39-s − 0.156·41-s − 0.609·43-s − 0.291·47-s − 3/7·49-s + 0.280·51-s + 0.549·53-s − 1.05·57-s + 1.56·59-s + 0.768·61-s + 0.251·63-s − 1.95·67-s + 0.481·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 196800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 41 | \( 1 + T \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 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.ae |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + 16 T + p T^{2} \) | 1.67.q |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 14 T + p T^{2} \) | 1.79.ao |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 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
−13.28728332534507, −12.89201010809002, −12.43687608335651, −12.02170175165095, −11.38758542800907, −10.89731876828780, −10.39139707625394, −10.13206536811288, −9.609265803057849, −8.852883315134067, −8.570822617446033, −8.016908726631564, −7.789955143306350, −7.028242958044707, −6.815955816137847, −6.010746112315324, −5.388393481191254, −4.898824136477226, −4.572645634734714, −3.987836873779235, −3.166127556025409, −2.712538254351731, −2.201333953867411, −1.711096912761761, −0.8005918400602032, 0,
0.8005918400602032, 1.711096912761761, 2.201333953867411, 2.712538254351731, 3.166127556025409, 3.987836873779235, 4.572645634734714, 4.898824136477226, 5.388393481191254, 6.010746112315324, 6.815955816137847, 7.028242958044707, 7.789955143306350, 8.016908726631564, 8.570822617446033, 8.852883315134067, 9.609265803057849, 10.13206536811288, 10.39139707625394, 10.89731876828780, 11.38758542800907, 12.02170175165095, 12.43687608335651, 12.89201010809002, 13.28728332534507
