| L(s) = 1 | − 2·2-s + 2·4-s + 3·5-s − 5·7-s − 6·10-s − 11-s − 2·13-s + 10·14-s − 4·16-s + 17-s + 6·20-s + 2·22-s + 4·23-s + 4·25-s + 4·26-s − 10·28-s − 2·29-s + 6·31-s + 8·32-s − 2·34-s − 15·35-s − 43-s − 2·44-s − 8·46-s + 9·47-s + 18·49-s − 8·50-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s + 1.34·5-s − 1.88·7-s − 1.89·10-s − 0.301·11-s − 0.554·13-s + 2.67·14-s − 16-s + 0.242·17-s + 1.34·20-s + 0.426·22-s + 0.834·23-s + 4/5·25-s + 0.784·26-s − 1.88·28-s − 0.371·29-s + 1.07·31-s + 1.41·32-s − 0.342·34-s − 2.53·35-s − 0.152·43-s − 0.301·44-s − 1.17·46-s + 1.31·47-s + 18/7·49-s − 1.13·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3249 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) | |
| 19 | \( 1 \) | |
| good | 2 | \( 1 + p T + p T^{2} \) | 1.2.c |
| 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 7 | \( 1 + 5 T + p T^{2} \) | 1.7.f |
| 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - T + p T^{2} \) | 1.17.ab |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 + p T^{2} \) | 1.37.a |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 - 9 T + p T^{2} \) | 1.47.aj |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
| 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
−8.680448170254320431881606216967, −7.42573159443448293402645158360, −7.00357033526907285371973300329, −6.15488117667979991166374049282, −5.61647921542357100461421630772, −4.40039433035204962000756580058, −3.00491682168355671002496814154, −2.44349751141625690337537557236, −1.22456142142336429144535327783, 0,
1.22456142142336429144535327783, 2.44349751141625690337537557236, 3.00491682168355671002496814154, 4.40039433035204962000756580058, 5.61647921542357100461421630772, 6.15488117667979991166374049282, 7.00357033526907285371973300329, 7.42573159443448293402645158360, 8.680448170254320431881606216967
