| L(s) = 1 | + 7-s − 3·11-s + 13-s − 3·17-s + 4·19-s − 9·23-s − 6·29-s + 2·31-s − 37-s + 3·41-s + 2·43-s − 6·47-s − 6·49-s − 9·53-s − 12·59-s − 5·61-s − 4·67-s − 9·71-s − 14·73-s − 3·77-s − 7·79-s − 15·89-s + 91-s − 5·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 0.377·7-s − 0.904·11-s + 0.277·13-s − 0.727·17-s + 0.917·19-s − 1.87·23-s − 1.11·29-s + 0.359·31-s − 0.164·37-s + 0.468·41-s + 0.304·43-s − 0.875·47-s − 6/7·49-s − 1.23·53-s − 1.56·59-s − 0.640·61-s − 0.488·67-s − 1.06·71-s − 1.63·73-s − 0.341·77-s − 0.787·79-s − 1.58·89-s + 0.104·91-s − 0.507·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 187200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 9 T + p T^{2} \) | 1.23.j |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + T + p T^{2} \) | 1.37.b |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 5 T + p T^{2} \) | 1.61.f |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 9 T + p T^{2} \) | 1.71.j |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 + 7 T + p T^{2} \) | 1.79.h |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 15 T + p T^{2} \) | 1.89.p |
| 97 | \( 1 + 5 T + p T^{2} \) | 1.97.f |
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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
−13.63643200759288, −13.15564244738941, −12.67198083400246, −12.20540010059097, −11.59974813356758, −11.33702786002824, −10.74273559644931, −10.38284545208021, −9.756027488492784, −9.420408284739267, −8.858196121218871, −8.158646225498224, −7.938645747346069, −7.479770487127216, −6.906815280062250, −6.189828426582956, −5.841284995826144, −5.356131257029928, −4.650840480934249, −4.326475252216724, −3.663949650289586, −2.962065698049353, −2.571865533111775, −1.606978955167296, −1.493967282184031, 0, 0,
1.493967282184031, 1.606978955167296, 2.571865533111775, 2.962065698049353, 3.663949650289586, 4.326475252216724, 4.650840480934249, 5.356131257029928, 5.841284995826144, 6.189828426582956, 6.906815280062250, 7.479770487127216, 7.938645747346069, 8.158646225498224, 8.858196121218871, 9.420408284739267, 9.756027488492784, 10.38284545208021, 10.74273559644931, 11.33702786002824, 11.59974813356758, 12.20540010059097, 12.67198083400246, 13.15564244738941, 13.63643200759288
