| L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s + 7-s + 8-s + 9-s − 10-s − 2·11-s − 12-s + 13-s + 14-s + 15-s + 16-s + 6·17-s + 18-s − 2·19-s − 20-s − 21-s − 2·22-s − 4·23-s − 24-s + 25-s + 26-s − 27-s + 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.603·11-s − 0.288·12-s + 0.277·13-s + 0.267·14-s + 0.258·15-s + 1/4·16-s + 1.45·17-s + 0.235·18-s − 0.458·19-s − 0.223·20-s − 0.218·21-s − 0.426·22-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.272980963\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.272980963\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 13 | \( 1 - T \) | |
| good | 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
| 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
−8.628061145911597081404836505184, −7.85249275293685041760385460781, −7.35208362597312479853045066833, −6.29912454633333280804413074205, −5.69876426716871714388295941868, −4.93221094181939303672799430536, −4.16767329291848311362731579414, −3.32324871579854953106753041219, −2.21014379877265652501289721456, −0.887609817770847120460902343637,
0.887609817770847120460902343637, 2.21014379877265652501289721456, 3.32324871579854953106753041219, 4.16767329291848311362731579414, 4.93221094181939303672799430536, 5.69876426716871714388295941868, 6.29912454633333280804413074205, 7.35208362597312479853045066833, 7.85249275293685041760385460781, 8.628061145911597081404836505184
