| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 7-s − 8-s + 9-s − 12-s + 13-s + 14-s + 16-s − 2·17-s − 18-s + 4·19-s + 21-s − 8·23-s + 24-s − 26-s − 27-s − 28-s − 6·29-s + 4·31-s − 32-s + 2·34-s + 36-s + 6·37-s − 4·38-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.288·12-s + 0.277·13-s + 0.267·14-s + 1/4·16-s − 0.485·17-s − 0.235·18-s + 0.917·19-s + 0.218·21-s − 1.66·23-s + 0.204·24-s − 0.196·26-s − 0.192·27-s − 0.188·28-s − 1.11·29-s + 0.718·31-s − 0.176·32-s + 0.342·34-s + 1/6·36-s + 0.986·37-s − 0.648·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7299021550\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7299021550\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 13 | \( 1 - T \) | |
| good | 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−16.30886177330289, −15.84985063939070, −15.24491439799068, −14.66208232591728, −13.86354446439775, −13.30986816408311, −12.79193244947310, −11.93915076413848, −11.64356722670253, −11.12650860298209, −10.31250454008218, −9.882943687823848, −9.466418496396353, −8.610135411600606, −8.066132244455712, −7.432733910443019, −6.725647607235549, −6.209319759702457, −5.604349497345899, −4.844275137885860, −3.944291005506533, −3.289154535766956, −2.272620366839011, −1.505175780038152, −0.4441531790969209,
0.4441531790969209, 1.505175780038152, 2.272620366839011, 3.289154535766956, 3.944291005506533, 4.844275137885860, 5.604349497345899, 6.209319759702457, 6.725647607235549, 7.432733910443019, 8.066132244455712, 8.610135411600606, 9.466418496396353, 9.882943687823848, 10.31250454008218, 11.12650860298209, 11.64356722670253, 11.93915076413848, 12.79193244947310, 13.30986816408311, 13.86354446439775, 14.66208232591728, 15.24491439799068, 15.84985063939070, 16.30886177330289
