| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 2·7-s − 8-s + 9-s + 10-s − 11-s − 12-s − 2·13-s + 2·14-s + 15-s + 16-s − 18-s − 8·19-s − 20-s + 2·21-s + 22-s − 4·23-s + 24-s + 25-s + 2·26-s − 27-s − 2·28-s + 6·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.301·11-s − 0.288·12-s − 0.554·13-s + 0.534·14-s + 0.258·15-s + 1/4·16-s − 0.235·18-s − 1.83·19-s − 0.223·20-s + 0.436·21-s + 0.213·22-s − 0.834·23-s + 0.204·24-s + 1/5·25-s + 0.392·26-s − 0.192·27-s − 0.377·28-s + 1.11·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 317130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 317130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8801134966\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8801134966\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 11 | \( 1 + T \) | |
| 31 | \( 1 \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 - 14 T + p T^{2} \) | 1.71.ao |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−12.47967337989589, −12.16787659015953, −11.78389997079950, −11.07657673467313, −10.82836821484524, −10.31870349893754, −9.976263673836031, −9.551684404855742, −8.842720656834594, −8.641633745263157, −7.891811386043379, −7.672006351119893, −7.051915694750374, −6.495991544612302, −6.235013707650483, −5.828584751258837, −4.904992829612950, −4.709510884096119, −3.900578728456986, −3.597298278842271, −2.701518317423445, −2.332679823759675, −1.753843976661837, −0.6920158927147031, −0.4353703950742165,
0.4353703950742165, 0.6920158927147031, 1.753843976661837, 2.332679823759675, 2.701518317423445, 3.597298278842271, 3.900578728456986, 4.709510884096119, 4.904992829612950, 5.828584751258837, 6.235013707650483, 6.495991544612302, 7.051915694750374, 7.672006351119893, 7.891811386043379, 8.641633745263157, 8.842720656834594, 9.551684404855742, 9.976263673836031, 10.31870349893754, 10.82836821484524, 11.07657673467313, 11.78389997079950, 12.16787659015953, 12.47967337989589
