| L(s) = 1 | + 2-s + 4-s + 3·5-s − 7-s + 8-s + 3·10-s − 14-s + 16-s + 17-s − 2·19-s + 3·20-s − 6·23-s + 4·25-s − 28-s + 3·29-s + 4·31-s + 32-s + 34-s − 3·35-s − 2·37-s − 2·38-s + 3·40-s + 8·43-s − 6·46-s + 9·47-s + 49-s + 4·50-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.34·5-s − 0.377·7-s + 0.353·8-s + 0.948·10-s − 0.267·14-s + 1/4·16-s + 0.242·17-s − 0.458·19-s + 0.670·20-s − 1.25·23-s + 4/5·25-s − 0.188·28-s + 0.557·29-s + 0.718·31-s + 0.176·32-s + 0.171·34-s − 0.507·35-s − 0.328·37-s − 0.324·38-s + 0.474·40-s + 1.21·43-s − 0.884·46-s + 1.31·47-s + 1/7·49-s + 0.565·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 361998 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 361998 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.367259133\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.367259133\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| 17 | \( 1 - T \) | |
| good | 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 9 T + p T^{2} \) | 1.47.aj |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 + 5 T + p T^{2} \) | 1.67.f |
| 71 | \( 1 - 15 T + p T^{2} \) | 1.71.ap |
| 73 | \( 1 - 16 T + p T^{2} \) | 1.73.aq |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 9 T + p T^{2} \) | 1.89.aj |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−12.51719830863342, −12.14223442608984, −11.90360515137829, −11.02254470315923, −10.72426128995315, −10.28725607614561, −9.821889280699328, −9.497159157075899, −8.965485993112536, −8.397082491083645, −7.914959973244781, −7.339030803263214, −6.803124534264776, −6.247091698750024, −6.063944386668934, −5.604510368193926, −5.039234276626044, −4.590828560764666, −3.865224550442342, −3.614298130431077, −2.664513991698848, −2.438333440203045, −1.943563466994187, −1.223099409315580, −0.5681160269487920,
0.5681160269487920, 1.223099409315580, 1.943563466994187, 2.438333440203045, 2.664513991698848, 3.614298130431077, 3.865224550442342, 4.590828560764666, 5.039234276626044, 5.604510368193926, 6.063944386668934, 6.247091698750024, 6.803124534264776, 7.339030803263214, 7.914959973244781, 8.397082491083645, 8.965485993112536, 9.497159157075899, 9.821889280699328, 10.28725607614561, 10.72426128995315, 11.02254470315923, 11.90360515137829, 12.14223442608984, 12.51719830863342
