| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 7-s − 8-s + 9-s − 2·11-s − 12-s + 14-s + 16-s + 5·17-s − 18-s − 2·19-s + 21-s + 2·22-s + 7·23-s + 24-s − 27-s − 28-s + 10·29-s − 10·31-s − 32-s + 2·33-s − 5·34-s + 36-s + 2·37-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 − 0.603·11-s − 0.288·12-s + 0.267·14-s + 1/4·16-s + 1.21·17-s − 0.235·18-s − 0.458·19-s + 0.218·21-s + 0.426·22-s + 1.45·23-s + 0.204·24-s − 0.192·27-s − 0.188·28-s + 1.85·29-s − 1.79·31-s − 0.176·32-s + 0.348·33-s − 0.857·34-s + 1/6·36-s + 0.328·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 17 | \( 1 - 5 T + p T^{2} \) | 1.17.af |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 7 T + p T^{2} \) | 1.23.ah |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 12 T + p T^{2} \) | 1.41.am |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 7 T + p T^{2} \) | 1.47.h |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 - 9 T + p T^{2} \) | 1.59.aj |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + 5 T + p T^{2} \) | 1.67.f |
| 71 | \( 1 + 2 T + p T^{2} \) | 1.71.c |
| 73 | \( 1 - 9 T + p T^{2} \) | 1.73.aj |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 8 T + p T^{2} \) | 1.89.ai |
| 97 | \( 1 - 13 T + p T^{2} \) | 1.97.an |
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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
−13.05026201633635, −12.84613831694713, −12.56846853969539, −11.93738049684658, −11.41093099479350, −10.95438617817541, −10.51562278768349, −10.26532915790157, −9.507730970412027, −9.260775940466758, −8.745397160914334, −7.991216785633071, −7.692513661935463, −7.226216866726943, −6.608739074500955, −6.187664859707560, −5.663109609827721, −5.126695165247606, −4.639437505054290, −3.907164131128969, −3.196904153374214, −2.791319275601007, −2.112973329915562, −1.225126493293934, −0.8179232705642690, 0,
0.8179232705642690, 1.225126493293934, 2.112973329915562, 2.791319275601007, 3.196904153374214, 3.907164131128969, 4.639437505054290, 5.126695165247606, 5.663109609827721, 6.187664859707560, 6.608739074500955, 7.226216866726943, 7.692513661935463, 7.991216785633071, 8.745397160914334, 9.260775940466758, 9.507730970412027, 10.26532915790157, 10.51562278768349, 10.95438617817541, 11.41093099479350, 11.93738049684658, 12.56846853969539, 12.84613831694713, 13.05026201633635
