| L(s) = 1 | + 2-s − 3-s − 4-s − 6-s − 3·8-s + 9-s + 4·11-s + 12-s + 2·13-s − 16-s + 18-s + 4·19-s + 4·22-s + 3·24-s + 2·26-s − 27-s + 2·29-s + 5·32-s − 4·33-s − 36-s − 10·37-s + 4·38-s − 2·39-s − 10·41-s − 4·43-s − 4·44-s − 8·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.408·6-s − 1.06·8-s + 1/3·9-s + 1.20·11-s + 0.288·12-s + 0.554·13-s − 1/4·16-s + 0.235·18-s + 0.917·19-s + 0.852·22-s + 0.612·24-s + 0.392·26-s − 0.192·27-s + 0.371·29-s + 0.883·32-s − 0.696·33-s − 1/6·36-s − 1.64·37-s + 0.648·38-s − 0.320·39-s − 1.56·41-s − 0.609·43-s − 0.603·44-s − 1.16·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 17 | \( 1 \) | |
| good | 2 | \( 1 - T + p T^{2} \) | 1.2.ab |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 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.ac |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−15.79808733652950, −15.15916494515028, −14.72198242693331, −14.03431847450384, −13.70478976870578, −13.22250086926941, −12.50490113665343, −11.99474381433688, −11.69061024327973, −11.10801141090013, −10.24252832832036, −9.810857423433633, −9.168696354992937, −8.609806086799086, −8.104875420158932, −7.027064678144183, −6.666718117023006, −6.036755439123625, −5.362021528899911, −4.915930571037800, −4.222894055903174, −3.509425130450013, −3.180749275302384, −1.809693898840095, −1.081384441976826, 0,
1.081384441976826, 1.809693898840095, 3.180749275302384, 3.509425130450013, 4.222894055903174, 4.915930571037800, 5.362021528899911, 6.036755439123625, 6.666718117023006, 7.027064678144183, 8.104875420158932, 8.609806086799086, 9.168696354992937, 9.810857423433633, 10.24252832832036, 11.10801141090013, 11.69061024327973, 11.99474381433688, 12.50490113665343, 13.22250086926941, 13.70478976870578, 14.03431847450384, 14.72198242693331, 15.15916494515028, 15.79808733652950
