| L(s) = 1 | − 3-s + 7-s + 9-s + 4·11-s + 2·13-s − 2·17-s + 4·19-s − 21-s − 27-s − 2·29-s + 8·31-s − 4·33-s + 2·37-s − 2·39-s + 2·41-s − 4·43-s + 49-s + 2·51-s + 10·53-s − 4·57-s − 12·59-s + 6·61-s + 63-s − 12·67-s + 6·73-s + 4·77-s − 8·79-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.377·7-s + 1/3·9-s + 1.20·11-s + 0.554·13-s − 0.485·17-s + 0.917·19-s − 0.218·21-s − 0.192·27-s − 0.371·29-s + 1.43·31-s − 0.696·33-s + 0.328·37-s − 0.320·39-s + 0.312·41-s − 0.609·43-s + 1/7·49-s + 0.280·51-s + 1.37·53-s − 0.529·57-s − 1.56·59-s + 0.768·61-s + 0.125·63-s − 1.46·67-s + 0.702·73-s + 0.455·77-s − 0.900·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.904176821\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.904176821\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 3 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| good | 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−8.447651848867857451610060749314, −7.60567042410819730199815007251, −6.83559751008852263080440304793, −6.22412736383067126183911262131, −5.51765229233286217889543043612, −4.59673899597302331020253987034, −3.99676058459627921033697673643, −2.99780568106058195411638741515, −1.72024847621442860075788489280, −0.865039611474349503579115946096,
0.865039611474349503579115946096, 1.72024847621442860075788489280, 2.99780568106058195411638741515, 3.99676058459627921033697673643, 4.59673899597302331020253987034, 5.51765229233286217889543043612, 6.22412736383067126183911262131, 6.83559751008852263080440304793, 7.60567042410819730199815007251, 8.447651848867857451610060749314
