| L(s) = 1 | + 2-s + 4-s + 4·5-s + 7-s + 8-s + 4·10-s − 13-s + 14-s + 16-s − 2·17-s + 7·19-s + 4·20-s − 6·23-s + 11·25-s − 26-s + 28-s − 3·29-s − 6·31-s + 32-s − 2·34-s + 4·35-s − 2·37-s + 7·38-s + 4·40-s − 5·41-s + 4·43-s − 6·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.78·5-s + 0.377·7-s + 0.353·8-s + 1.26·10-s − 0.277·13-s + 0.267·14-s + 1/4·16-s − 0.485·17-s + 1.60·19-s + 0.894·20-s − 1.25·23-s + 11/5·25-s − 0.196·26-s + 0.188·28-s − 0.557·29-s − 1.07·31-s + 0.176·32-s − 0.342·34-s + 0.676·35-s − 0.328·37-s + 1.13·38-s + 0.632·40-s − 0.780·41-s + 0.609·43-s − 0.884·46-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 | \( 1 - 4 T + p T^{2} \) | 1.5.ae |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 7 T + p T^{2} \) | 1.19.ah |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 5 T + p T^{2} \) | 1.41.f |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 7 T + p T^{2} \) | 1.47.h |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 - 2 T + p T^{2} \) | 1.59.ac |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 - 15 T + p T^{2} \) | 1.67.ap |
| 71 | \( 1 + 13 T + p T^{2} \) | 1.71.n |
| 73 | \( 1 + 8 T + p T^{2} \) | 1.73.i |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 - 2 T + p T^{2} \) | 1.83.ac |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 + 16 T + p T^{2} \) | 1.97.q |
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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
−13.29325526569779, −13.07389203179945, −12.43459810780550, −12.00639147678146, −11.42846892745092, −11.02345732216476, −10.50650772251860, −9.883225498843216, −9.662210422450270, −9.288647353218750, −8.502796868962555, −8.141281384446139, −7.232511354079111, −7.114920008289858, −6.401455125296612, −5.885061856623901, −5.466207476294847, −5.222325018160381, −4.578580466394056, −3.953863117834468, −3.253969935438452, −2.750100236217631, −2.086372275908229, −1.693633332448824, −1.194454651556501, 0,
1.194454651556501, 1.693633332448824, 2.086372275908229, 2.750100236217631, 3.253969935438452, 3.953863117834468, 4.578580466394056, 5.222325018160381, 5.466207476294847, 5.885061856623901, 6.401455125296612, 7.114920008289858, 7.232511354079111, 8.141281384446139, 8.502796868962555, 9.288647353218750, 9.662210422450270, 9.883225498843216, 10.50650772251860, 11.02345732216476, 11.42846892745092, 12.00639147678146, 12.43459810780550, 13.07389203179945, 13.29325526569779
