| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 8-s + 9-s + 4·11-s − 12-s + 13-s + 16-s + 6·17-s + 18-s − 4·19-s + 4·22-s + 4·23-s − 24-s + 26-s − 27-s + 2·29-s + 4·31-s + 32-s − 4·33-s + 6·34-s + 36-s + 10·37-s − 4·38-s − 39-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.353·8-s + 1/3·9-s + 1.20·11-s − 0.288·12-s + 0.277·13-s + 1/4·16-s + 1.45·17-s + 0.235·18-s − 0.917·19-s + 0.852·22-s + 0.834·23-s − 0.204·24-s + 0.196·26-s − 0.192·27-s + 0.371·29-s + 0.718·31-s + 0.176·32-s − 0.696·33-s + 1.02·34-s + 1/6·36-s + 1.64·37-s − 0.648·38-s − 0.160·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 95550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 95550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.222917051\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.222917051\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 3 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 16 T + p T^{2} \) | 1.71.aq |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 16 T + p T^{2} \) | 1.79.aq |
| 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.c |
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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
−13.71915261648579, −13.36952543631633, −12.66366044476730, −12.37622710560358, −11.92688390884453, −11.42437642734453, −10.95852739231854, −10.57137091168629, −9.734431530512598, −9.596038951304247, −8.835245534111639, −8.046664398200418, −7.886734755094118, −6.863284033059460, −6.657512142319874, −6.186907504599913, −5.571175062936742, −5.065134560684121, −4.494988120281985, −3.871158626763223, −3.511727583087957, −2.702086999289321, −2.028261063123546, −1.102362433690131, −0.8039715926571409,
0.8039715926571409, 1.102362433690131, 2.028261063123546, 2.702086999289321, 3.511727583087957, 3.871158626763223, 4.494988120281985, 5.065134560684121, 5.571175062936742, 6.186907504599913, 6.657512142319874, 6.863284033059460, 7.886734755094118, 8.046664398200418, 8.835245534111639, 9.596038951304247, 9.734431530512598, 10.57137091168629, 10.95852739231854, 11.42437642734453, 11.92688390884453, 12.37622710560358, 12.66366044476730, 13.36952543631633, 13.71915261648579
