| L(s) = 1 | + 2-s + 4-s + 2·5-s − 7-s + 8-s + 2·10-s + 13-s − 14-s + 16-s − 2·17-s + 4·19-s + 2·20-s + 4·23-s − 25-s + 26-s − 28-s − 6·29-s + 32-s − 2·34-s − 2·35-s − 10·37-s + 4·38-s + 2·40-s − 2·41-s + 4·43-s + 4·46-s + 49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.894·5-s − 0.377·7-s + 0.353·8-s + 0.632·10-s + 0.277·13-s − 0.267·14-s + 1/4·16-s − 0.485·17-s + 0.917·19-s + 0.447·20-s + 0.834·23-s − 1/5·25-s + 0.196·26-s − 0.188·28-s − 1.11·29-s + 0.176·32-s − 0.342·34-s − 0.338·35-s − 1.64·37-s + 0.648·38-s + 0.316·40-s − 0.312·41-s + 0.609·43-s + 0.589·46-s + 1/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 198198 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 198198 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 3 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + 16 T + p T^{2} \) | 1.71.q |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 8 T + p T^{2} \) | 1.83.ai |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−13.44352735828975, −12.94656872138352, −12.42608430628635, −12.01750701908848, −11.46902017724031, −11.00803218531451, −10.45986837645621, −10.17104951226104, −9.484399671903113, −9.022773578027792, −8.855796842646167, −7.825242272472447, −7.538240460727910, −6.911566524901962, −6.494068001922419, −5.933269470669488, −5.559429729564116, −5.075051362355731, −4.544127670778898, −3.845759189172308, −3.297803726711147, −2.923875606342291, −2.084738315328208, −1.736312974147092, −0.9880899397354454, 0,
0.9880899397354454, 1.736312974147092, 2.084738315328208, 2.923875606342291, 3.297803726711147, 3.845759189172308, 4.544127670778898, 5.075051362355731, 5.559429729564116, 5.933269470669488, 6.494068001922419, 6.911566524901962, 7.538240460727910, 7.825242272472447, 8.855796842646167, 9.022773578027792, 9.484399671903113, 10.17104951226104, 10.45986837645621, 11.00803218531451, 11.46902017724031, 12.01750701908848, 12.42608430628635, 12.94656872138352, 13.44352735828975
