| 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 − 3·17-s − 18-s + 2·19-s + 21-s + 3·23-s − 24-s − 5·25-s − 26-s + 27-s + 28-s − 10·31-s − 32-s + 3·34-s + 36-s + 8·37-s − 2·38-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 − 0.727·17-s − 0.235·18-s + 0.458·19-s + 0.218·21-s + 0.625·23-s − 0.204·24-s − 25-s − 0.196·26-s + 0.192·27-s + 0.188·28-s − 1.79·31-s − 0.176·32-s + 0.514·34-s + 1/6·36-s + 1.31·37-s − 0.324·38-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + p T^{2} \) | 1.5.a |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 + 7 T + p T^{2} \) | 1.73.h |
| 79 | \( 1 - 14 T + p T^{2} \) | 1.79.ao |
| 83 | \( 1 - 9 T + p T^{2} \) | 1.83.aj |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 7 T + p T^{2} \) | 1.97.h |
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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
−14.60138245294390, −13.98049747290535, −13.38256771773690, −13.07923850497786, −12.47118630261669, −11.68690423009972, −11.46080478175513, −10.84088867959988, −10.40739893376865, −9.710240140273846, −9.311729083824786, −8.875266025668697, −8.333085459242417, −7.765612424118930, −7.437006416848385, −6.734661873352709, −6.293020853091954, −5.458876774998943, −5.032184176145921, −4.146607385288870, −3.661111349992157, −2.984598781211535, −2.219758340554255, −1.757356740843292, −0.9802073164676437, 0,
0.9802073164676437, 1.757356740843292, 2.219758340554255, 2.984598781211535, 3.661111349992157, 4.146607385288870, 5.032184176145921, 5.458876774998943, 6.293020853091954, 6.734661873352709, 7.437006416848385, 7.765612424118930, 8.333085459242417, 8.875266025668697, 9.311729083824786, 9.710240140273846, 10.40739893376865, 10.84088867959988, 11.46080478175513, 11.68690423009972, 12.47118630261669, 13.07923850497786, 13.38256771773690, 13.98049747290535, 14.60138245294390
