| L(s) = 1 | + 2·3-s + 9-s − 2·11-s − 13-s − 2·17-s − 2·19-s + 2·23-s − 4·27-s − 6·29-s − 2·31-s − 4·33-s + 6·37-s − 2·39-s + 2·41-s + 6·43-s − 8·47-s − 7·49-s − 4·51-s + 2·53-s − 4·57-s − 6·59-s − 14·61-s + 4·69-s − 10·71-s + 2·73-s + 4·79-s − 11·81-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1/3·9-s − 0.603·11-s − 0.277·13-s − 0.485·17-s − 0.458·19-s + 0.417·23-s − 0.769·27-s − 1.11·29-s − 0.359·31-s − 0.696·33-s + 0.986·37-s − 0.320·39-s + 0.312·41-s + 0.914·43-s − 1.16·47-s − 49-s − 0.560·51-s + 0.274·53-s − 0.529·57-s − 0.781·59-s − 1.79·61-s + 0.481·69-s − 1.18·71-s + 0.234·73-s + 0.450·79-s − 1.22·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 + 10 T + p T^{2} \) | 1.71.k |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
| 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
−7.85052658969108827311113346194, −7.44305818517757472018800684692, −6.47202411196321581767915802789, −5.68989193388888299158530738836, −4.78903400360447838269387412877, −4.01501047657526438733304959362, −3.12609814824846506170449543026, −2.50757282461396971152814594714, −1.65667165457222088341231259370, 0,
1.65667165457222088341231259370, 2.50757282461396971152814594714, 3.12609814824846506170449543026, 4.01501047657526438733304959362, 4.78903400360447838269387412877, 5.68989193388888299158530738836, 6.47202411196321581767915802789, 7.44305818517757472018800684692, 7.85052658969108827311113346194
