| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s − 8-s + 9-s + 10-s + 12-s + 13-s − 15-s + 16-s + 4·17-s − 18-s + 6·19-s − 20-s − 24-s + 25-s − 26-s + 27-s + 6·29-s + 30-s + 2·31-s − 32-s − 4·34-s + 36-s + 4·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.288·12-s + 0.277·13-s − 0.258·15-s + 1/4·16-s + 0.970·17-s − 0.235·18-s + 1.37·19-s − 0.223·20-s − 0.204·24-s + 1/5·25-s − 0.196·26-s + 0.192·27-s + 1.11·29-s + 0.182·30-s + 0.359·31-s − 0.176·32-s − 0.685·34-s + 1/6·36-s + 0.657·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.332268693\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.332268693\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−15.76590373037474, −15.34398417147601, −14.65482828210726, −14.05163547008752, −13.79299598771016, −12.86293940864128, −12.30469289938020, −11.91701558261291, −11.16072088498975, −10.72999397173804, −9.939806785173112, −9.561238178358686, −9.021619027851031, −8.241138077272861, −7.908569762731420, −7.360341002989877, −6.768149123038685, −5.938244978699768, −5.346783869414042, −4.417050551122865, −3.757295531641517, −2.970251830347377, −2.519222266765214, −1.292543823209795, −0.7918897227693301,
0.7918897227693301, 1.292543823209795, 2.519222266765214, 2.970251830347377, 3.757295531641517, 4.417050551122865, 5.346783869414042, 5.938244978699768, 6.768149123038685, 7.360341002989877, 7.908569762731420, 8.241138077272861, 9.021619027851031, 9.561238178358686, 9.939806785173112, 10.72999397173804, 11.16072088498975, 11.91701558261291, 12.30469289938020, 12.86293940864128, 13.79299598771016, 14.05163547008752, 14.65482828210726, 15.34398417147601, 15.76590373037474
