| L(s) = 1 | + 2·3-s + 9-s + 2·11-s − 6·13-s + 17-s − 8·19-s − 2·23-s − 4·27-s + 6·29-s − 2·31-s + 4·33-s − 6·37-s − 12·39-s − 2·41-s − 4·43-s − 4·47-s + 2·51-s + 10·53-s − 16·57-s + 10·61-s + 8·67-s − 4·69-s − 14·71-s + 10·73-s + 14·79-s − 11·81-s + 4·83-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1/3·9-s + 0.603·11-s − 1.66·13-s + 0.242·17-s − 1.83·19-s − 0.417·23-s − 0.769·27-s + 1.11·29-s − 0.359·31-s + 0.696·33-s − 0.986·37-s − 1.92·39-s − 0.312·41-s − 0.609·43-s − 0.583·47-s + 0.280·51-s + 1.37·53-s − 2.11·57-s + 1.28·61-s + 0.977·67-s − 0.481·69-s − 1.66·71-s + 1.17·73-s + 1.57·79-s − 1.22·81-s + 0.439·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 333200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 333200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.366083368\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.366083368\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| 17 | \( 1 - T \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + 14 T + p T^{2} \) | 1.71.o |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - 14 T + p T^{2} \) | 1.79.ao |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−12.52016705443524, −12.16974243563317, −11.90664043170505, −11.20409116768669, −10.69795315015382, −10.12019534128343, −9.871970885344010, −9.374516781239867, −8.836498310595432, −8.458063140613392, −8.140008599437484, −7.626257648548887, −6.958499411685375, −6.741066923797094, −6.189361600710148, −5.377375627734027, −5.068001796704064, −4.353904728626786, −3.957062003123180, −3.469516810173619, −2.802677755151505, −2.283285042112391, −2.056047262490559, −1.266443309835458, −0.2657414545036209,
0.2657414545036209, 1.266443309835458, 2.056047262490559, 2.283285042112391, 2.802677755151505, 3.469516810173619, 3.957062003123180, 4.353904728626786, 5.068001796704064, 5.377375627734027, 6.189361600710148, 6.741066923797094, 6.958499411685375, 7.626257648548887, 8.140008599437484, 8.458063140613392, 8.836498310595432, 9.374516781239867, 9.871970885344010, 10.12019534128343, 10.69795315015382, 11.20409116768669, 11.90664043170505, 12.16974243563317, 12.52016705443524
