| L(s) = 1 | + 2-s + 2·3-s + 4-s + 2·6-s + 8-s + 9-s + 3·11-s + 2·12-s − 4·13-s + 16-s − 4·17-s + 18-s + 4·19-s + 3·22-s − 4·23-s + 2·24-s − 4·26-s − 4·27-s − 9·29-s − 2·31-s + 32-s + 6·33-s − 4·34-s + 36-s − 3·37-s + 4·38-s − 8·39-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.816·6-s + 0.353·8-s + 1/3·9-s + 0.904·11-s + 0.577·12-s − 1.10·13-s + 1/4·16-s − 0.970·17-s + 0.235·18-s + 0.917·19-s + 0.639·22-s − 0.834·23-s + 0.408·24-s − 0.784·26-s − 0.769·27-s − 1.67·29-s − 0.359·31-s + 0.176·32-s + 1.04·33-s − 0.685·34-s + 1/6·36-s − 0.493·37-s + 0.648·38-s − 1.28·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 193550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 193550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.232765446\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.232765446\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| 79 | \( 1 + T \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 9 T + p T^{2} \) | 1.29.j |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 - 8 T + p T^{2} \) | 1.41.ai |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - T + p T^{2} \) | 1.47.ab |
| 53 | \( 1 - 11 T + p T^{2} \) | 1.53.al |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 - 3 T + p T^{2} \) | 1.61.ad |
| 67 | \( 1 - 7 T + p T^{2} \) | 1.67.ah |
| 71 | \( 1 + 15 T + p T^{2} \) | 1.71.p |
| 73 | \( 1 - 9 T + p T^{2} \) | 1.73.aj |
| 83 | \( 1 - 5 T + p T^{2} \) | 1.83.af |
| 89 | \( 1 - T + p T^{2} \) | 1.89.ab |
| 97 | \( 1 + 5 T + p T^{2} \) | 1.97.f |
| 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
−13.17602916642819, −12.74946275618063, −12.19325800651634, −11.78481750682173, −11.36145954902172, −10.80713705528219, −10.28285533712447, −9.512544194934157, −9.279755101520443, −9.046823891465648, −8.213744662764387, −7.802980529452229, −7.258864805721830, −7.040101356381109, −6.277777826141482, −5.658699482383378, −5.373644111422756, −4.506396811199757, −4.059495182298256, −3.740647563362672, −3.047921830034010, −2.506155766319403, −2.074180417335937, −1.525404888883299, −0.4874084887290898,
0.4874084887290898, 1.525404888883299, 2.074180417335937, 2.506155766319403, 3.047921830034010, 3.740647563362672, 4.059495182298256, 4.506396811199757, 5.373644111422756, 5.658699482383378, 6.277777826141482, 7.040101356381109, 7.258864805721830, 7.802980529452229, 8.213744662764387, 9.046823891465648, 9.279755101520443, 9.512544194934157, 10.28285533712447, 10.80713705528219, 11.36145954902172, 11.78481750682173, 12.19325800651634, 12.74946275618063, 13.17602916642819
