| L(s) = 1 | − 2·4-s + 2·7-s − 11-s + 4·16-s + 3·17-s + 4·19-s − 7·23-s − 4·28-s − 3·29-s + 8·31-s + 6·37-s − 8·41-s + 6·43-s + 2·44-s − 3·49-s − 8·53-s + 9·59-s + 6·61-s − 8·64-s − 67-s − 6·68-s − 13·71-s + 11·73-s − 8·76-s − 2·77-s + 10·79-s + 9·83-s + ⋯ |
| L(s) = 1 | − 4-s + 0.755·7-s − 0.301·11-s + 16-s + 0.727·17-s + 0.917·19-s − 1.45·23-s − 0.755·28-s − 0.557·29-s + 1.43·31-s + 0.986·37-s − 1.24·41-s + 0.914·43-s + 0.301·44-s − 3/7·49-s − 1.09·53-s + 1.17·59-s + 0.768·61-s − 64-s − 0.122·67-s − 0.727·68-s − 1.54·71-s + 1.28·73-s − 0.917·76-s − 0.227·77-s + 1.12·79-s + 0.987·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 165825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.151120898\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.151120898\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| 67 | \( 1 + T \) | |
| good | 2 | \( 1 + p T^{2} \) | 1.2.a |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 7 T + p T^{2} \) | 1.23.h |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 8 T + p T^{2} \) | 1.53.i |
| 59 | \( 1 - 9 T + p T^{2} \) | 1.59.aj |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 71 | \( 1 + 13 T + p T^{2} \) | 1.71.n |
| 73 | \( 1 - 11 T + p T^{2} \) | 1.73.al |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 - 9 T + p T^{2} \) | 1.83.aj |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + p T^{2} \) | 1.97.a |
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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
−13.30566710625526, −12.83873814184954, −12.23801774294862, −11.88916987538857, −11.43767498290149, −10.87788334178277, −10.14725541079315, −9.955978300360373, −9.515574945579397, −8.917330944905970, −8.300245438573519, −7.953908781632720, −7.732184874979574, −7.008231321614952, −6.228834948750947, −5.773078041870974, −5.303487330281137, −4.732739879924176, −4.415605147925750, −3.653073719175898, −3.298006111370422, −2.475503913381602, −1.790080289319179, −1.075059417574440, −0.4983020833954347,
0.4983020833954347, 1.075059417574440, 1.790080289319179, 2.475503913381602, 3.298006111370422, 3.653073719175898, 4.415605147925750, 4.732739879924176, 5.303487330281137, 5.773078041870974, 6.228834948750947, 7.008231321614952, 7.732184874979574, 7.953908781632720, 8.300245438573519, 8.917330944905970, 9.515574945579397, 9.955978300360373, 10.14725541079315, 10.87788334178277, 11.43767498290149, 11.88916987538857, 12.23801774294862, 12.83873814184954, 13.30566710625526
