| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s − 4·7-s + 8-s + 9-s + 10-s + 6·11-s − 12-s − 13-s − 4·14-s − 15-s + 16-s + 4·17-s + 18-s + 5·19-s + 20-s + 4·21-s + 6·22-s − 24-s + 25-s − 26-s − 27-s − 4·28-s + 2·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 1.51·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s + 1.80·11-s − 0.288·12-s − 0.277·13-s − 1.06·14-s − 0.258·15-s + 1/4·16-s + 0.970·17-s + 0.235·18-s + 1.14·19-s + 0.223·20-s + 0.872·21-s + 1.27·22-s − 0.204·24-s + 1/5·25-s − 0.196·26-s − 0.192·27-s − 0.755·28-s + 0.371·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 206310 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 206310 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.841045957\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.841045957\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 13 | \( 1 + T \) | |
| 23 | \( 1 \) | |
| good | 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 7 T + p T^{2} \) | 1.41.h |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 - 7 T + p T^{2} \) | 1.47.ah |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 15 T + p T^{2} \) | 1.71.ap |
| 73 | \( 1 + p T^{2} \) | 1.73.a |
| 79 | \( 1 - T + p T^{2} \) | 1.79.ab |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 3 T + p T^{2} \) | 1.89.ad |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.05916394177264, −12.39408722880379, −12.14053282568387, −11.87248628084041, −11.36820664042823, −10.68669935119552, −10.14864711741966, −9.758252920970767, −9.494287860139616, −8.940595815461981, −8.329192155542403, −7.461151036439240, −7.026831767190553, −6.763759823732942, −6.027639392327476, −6.014570477007664, −5.301618569243737, −4.802973073683896, −4.079825293349530, −3.510889514839096, −3.347498094269590, −2.559215766737712, −1.842151533719852, −1.052228333424256, −0.6646282368293576,
0.6646282368293576, 1.052228333424256, 1.842151533719852, 2.559215766737712, 3.347498094269590, 3.510889514839096, 4.079825293349530, 4.802973073683896, 5.301618569243737, 6.014570477007664, 6.027639392327476, 6.763759823732942, 7.026831767190553, 7.461151036439240, 8.329192155542403, 8.940595815461981, 9.494287860139616, 9.758252920970767, 10.14864711741966, 10.68669935119552, 11.36820664042823, 11.87248628084041, 12.14053282568387, 12.39408722880379, 13.05916394177264
