| L(s) = 1 | − 2-s + 4-s + 3·5-s − 2·7-s − 8-s − 3·10-s − 5·13-s + 2·14-s + 16-s + 3·17-s − 2·19-s + 3·20-s − 6·23-s + 4·25-s + 5·26-s − 2·28-s − 3·29-s + 2·31-s − 32-s − 3·34-s − 6·35-s − 7·37-s + 2·38-s − 3·40-s + 3·41-s − 8·43-s + 6·46-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.34·5-s − 0.755·7-s − 0.353·8-s − 0.948·10-s − 1.38·13-s + 0.534·14-s + 1/4·16-s + 0.727·17-s − 0.458·19-s + 0.670·20-s − 1.25·23-s + 4/5·25-s + 0.980·26-s − 0.377·28-s − 0.557·29-s + 0.359·31-s − 0.176·32-s − 0.514·34-s − 1.01·35-s − 1.15·37-s + 0.324·38-s − 0.474·40-s + 0.468·41-s − 1.21·43-s + 0.884·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2178 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2178 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 11 | \( 1 \) | |
| good | 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 13 | \( 1 + 5 T + p T^{2} \) | 1.13.f |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 10 T + p T^{2} \) | 1.67.k |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 + 2 T + p T^{2} \) | 1.79.c |
| 83 | \( 1 - 18 T + p T^{2} \) | 1.83.as |
| 89 | \( 1 - 9 T + p T^{2} \) | 1.89.aj |
| 97 | \( 1 - 11 T + p T^{2} \) | 1.97.al |
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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
−8.900098133636718958293788517920, −7.932254624876906788240570504173, −7.12971528935758372285009464422, −6.34068742561049769477086584673, −5.73995378988955117370178521927, −4.84624266923289247624863204778, −3.44890016858367005507255422381, −2.44680839597863177893297068489, −1.68556648322455773231137315655, 0,
1.68556648322455773231137315655, 2.44680839597863177893297068489, 3.44890016858367005507255422381, 4.84624266923289247624863204778, 5.73995378988955117370178521927, 6.34068742561049769477086584673, 7.12971528935758372285009464422, 7.932254624876906788240570504173, 8.900098133636718958293788517920
