| L(s) = 1 | + 2-s + 3-s − 4-s + 6-s + 2·7-s − 3·8-s + 9-s − 4·11-s − 12-s − 6·13-s + 2·14-s − 16-s − 4·17-s + 18-s − 2·19-s + 2·21-s − 4·22-s − 3·24-s − 5·25-s − 6·26-s + 27-s − 2·28-s + 2·29-s + 4·31-s + 5·32-s − 4·33-s − 4·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.408·6-s + 0.755·7-s − 1.06·8-s + 1/3·9-s − 1.20·11-s − 0.288·12-s − 1.66·13-s + 0.534·14-s − 1/4·16-s − 0.970·17-s + 0.235·18-s − 0.458·19-s + 0.436·21-s − 0.852·22-s − 0.612·24-s − 25-s − 1.17·26-s + 0.192·27-s − 0.377·28-s + 0.371·29-s + 0.718·31-s + 0.883·32-s − 0.696·33-s − 0.685·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1587 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1587 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 23 | \( 1 \) | |
| good | 2 | \( 1 - T + p T^{2} \) | 1.2.ab |
| 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 - 10 T + p T^{2} \) | 1.67.ak |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 16 T + p T^{2} \) | 1.89.aq |
| 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
−8.912115883457701282549736658807, −8.211605142746493961460634249227, −7.57852957473500430748876570413, −6.55154894173086314444633827884, −5.34423672570445956439168284893, −4.82108321928925973573258671004, −4.12133946477945840123408046984, −2.86791479523701443173120189217, −2.13959716896567237142416295040, 0,
2.13959716896567237142416295040, 2.86791479523701443173120189217, 4.12133946477945840123408046984, 4.82108321928925973573258671004, 5.34423672570445956439168284893, 6.55154894173086314444633827884, 7.57852957473500430748876570413, 8.211605142746493961460634249227, 8.912115883457701282549736658807
