| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 2·7-s − 8-s − 2·9-s + 3·11-s − 12-s − 4·13-s + 2·14-s + 16-s + 2·18-s + 5·19-s + 2·21-s − 3·22-s − 6·23-s + 24-s + 4·26-s + 5·27-s − 2·28-s − 2·31-s − 32-s − 3·33-s − 2·36-s − 2·37-s − 5·38-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.755·7-s − 0.353·8-s − 2/3·9-s + 0.904·11-s − 0.288·12-s − 1.10·13-s + 0.534·14-s + 1/4·16-s + 0.471·18-s + 1.14·19-s + 0.436·21-s − 0.639·22-s − 1.25·23-s + 0.204·24-s + 0.784·26-s + 0.962·27-s − 0.377·28-s − 0.359·31-s − 0.176·32-s − 0.522·33-s − 1/3·36-s − 0.328·37-s − 0.811·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5189037435\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5189037435\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 17 | \( 1 \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 13 T + p T^{2} \) | 1.67.n |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 + 9 T + p T^{2} \) | 1.83.j |
| 89 | \( 1 - 15 T + p T^{2} \) | 1.89.ap |
| 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
−16.38402961144619, −15.78879452251279, −15.02685686724262, −14.49403059289836, −13.95638300496924, −13.33196466351309, −12.37276852897552, −12.02218487990037, −11.73149971840249, −10.99529949274758, −10.21682996677420, −9.940395857459385, −9.118951017328723, −8.898043793872648, −7.901574712335349, −7.342781591127439, −6.790888734117873, −6.014542622781423, −5.696861489789325, −4.813827776493977, −3.925615894743918, −3.119820640697974, −2.449033427744365, −1.414985892350019, −0.3743007147376579,
0.3743007147376579, 1.414985892350019, 2.449033427744365, 3.119820640697974, 3.925615894743918, 4.813827776493977, 5.696861489789325, 6.014542622781423, 6.790888734117873, 7.342781591127439, 7.901574712335349, 8.898043793872648, 9.118951017328723, 9.940395857459385, 10.21682996677420, 10.99529949274758, 11.73149971840249, 12.02218487990037, 12.37276852897552, 13.33196466351309, 13.95638300496924, 14.49403059289836, 15.02685686724262, 15.78879452251279, 16.38402961144619
