| L(s) = 1 | − 2-s + 3-s + 4-s − 2·5-s − 6-s + 7-s − 8-s + 9-s + 2·10-s + 12-s − 13-s − 14-s − 2·15-s + 16-s − 18-s + 4·19-s − 2·20-s + 21-s − 4·23-s − 24-s − 25-s + 26-s + 27-s + 28-s + 8·29-s + 2·30-s + 4·31-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.894·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.632·10-s + 0.288·12-s − 0.277·13-s − 0.267·14-s − 0.516·15-s + 1/4·16-s − 0.235·18-s + 0.917·19-s − 0.447·20-s + 0.218·21-s − 0.834·23-s − 0.204·24-s − 1/5·25-s + 0.196·26-s + 0.192·27-s + 0.188·28-s + 1.48·29-s + 0.365·30-s + 0.718·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 66066 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 66066 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.261939456\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.261939456\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 7 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + p T^{2} \) | 1.37.a |
| 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.ae |
| 53 | \( 1 - 14 T + p T^{2} \) | 1.53.ao |
| 59 | \( 1 - 14 T + p T^{2} \) | 1.59.ao |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + 10 T + p T^{2} \) | 1.71.k |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - 6 T + p T^{2} \) | 1.79.ag |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + 4 T + p T^{2} \) | 1.97.e |
| 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
−14.19191187408959, −13.85774715002779, −13.26669193933436, −12.49792660569802, −12.06848608021219, −11.68115710230324, −11.27236371091692, −10.46883037806049, −10.14126727256489, −9.626436731517492, −8.973459564234662, −8.498156500670855, −8.051217643959731, −7.608749036588986, −7.191506893959023, −6.593417879253808, −5.828091239888572, −5.256348487871701, −4.394456548925539, −4.022033573594380, −3.356564209528683, −2.530655453770393, −2.229525366822332, −1.069941186411519, −0.6537238485427747,
0.6537238485427747, 1.069941186411519, 2.229525366822332, 2.530655453770393, 3.356564209528683, 4.022033573594380, 4.394456548925539, 5.256348487871701, 5.828091239888572, 6.593417879253808, 7.191506893959023, 7.608749036588986, 8.051217643959731, 8.498156500670855, 8.973459564234662, 9.626436731517492, 10.14126727256489, 10.46883037806049, 11.27236371091692, 11.68115710230324, 12.06848608021219, 12.49792660569802, 13.26669193933436, 13.85774715002779, 14.19191187408959
