| L(s) = 1 | + 2-s + 4-s + 2·7-s + 8-s + 4·11-s + 2·14-s + 16-s − 17-s + 19-s + 4·22-s + 4·23-s + 2·28-s − 6·29-s + 8·31-s + 32-s − 34-s − 2·37-s + 38-s + 10·43-s + 4·44-s + 4·46-s − 3·49-s + 2·53-s + 2·56-s − 6·58-s + 10·59-s + 2·61-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.755·7-s + 0.353·8-s + 1.20·11-s + 0.534·14-s + 1/4·16-s − 0.242·17-s + 0.229·19-s + 0.852·22-s + 0.834·23-s + 0.377·28-s − 1.11·29-s + 1.43·31-s + 0.176·32-s − 0.171·34-s − 0.328·37-s + 0.162·38-s + 1.52·43-s + 0.603·44-s + 0.589·46-s − 3/7·49-s + 0.274·53-s + 0.267·56-s − 0.787·58-s + 1.30·59-s + 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 145350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 145350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.742290779\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.742290779\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 17 | \( 1 + T \) | |
| 19 | \( 1 - T \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 10 T + p T^{2} \) | 1.59.ak |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 + 10 T + p T^{2} \) | 1.71.k |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 - 12 T + p T^{2} \) | 1.97.am |
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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
−13.47051322766143, −12.94989682614166, −12.38305950982527, −11.88689035455429, −11.57227823584505, −11.13934238172518, −10.65541655053989, −10.10618570022214, −9.462873347649929, −8.967980512428706, −8.633748571392481, −7.776467929308287, −7.578160573470848, −6.844366907114698, −6.458383555915946, −5.922495258358789, −5.340373115059717, −4.721412467661271, −4.461881190322308, −3.642059629868290, −3.422233370770314, −2.438555673913616, −2.047019455575084, −1.227255927335151, −0.7428532202094735,
0.7428532202094735, 1.227255927335151, 2.047019455575084, 2.438555673913616, 3.422233370770314, 3.642059629868290, 4.461881190322308, 4.721412467661271, 5.340373115059717, 5.922495258358789, 6.458383555915946, 6.844366907114698, 7.578160573470848, 7.776467929308287, 8.633748571392481, 8.967980512428706, 9.462873347649929, 10.10618570022214, 10.65541655053989, 11.13934238172518, 11.57227823584505, 11.88689035455429, 12.38305950982527, 12.94989682614166, 13.47051322766143
