| L(s) = 1 | − 2-s + 4-s − 2·7-s − 8-s + 3·11-s + 4·13-s + 2·14-s + 16-s − 3·17-s + 5·19-s − 3·22-s + 6·23-s − 4·26-s − 2·28-s + 2·31-s − 32-s + 3·34-s − 2·37-s − 5·38-s + 3·41-s + 4·43-s + 3·44-s − 6·46-s + 12·47-s − 3·49-s + 4·52-s + 6·53-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.755·7-s − 0.353·8-s + 0.904·11-s + 1.10·13-s + 0.534·14-s + 1/4·16-s − 0.727·17-s + 1.14·19-s − 0.639·22-s + 1.25·23-s − 0.784·26-s − 0.377·28-s + 0.359·31-s − 0.176·32-s + 0.514·34-s − 0.328·37-s − 0.811·38-s + 0.468·41-s + 0.609·43-s + 0.452·44-s − 0.884·46-s + 1.75·47-s − 3/7·49-s + 0.554·52-s + 0.824·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.031229189\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.031229189\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 13 T + p T^{2} \) | 1.67.an |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 + 9 T + p T^{2} \) | 1.83.j |
| 89 | \( 1 + 15 T + p T^{2} \) | 1.89.p |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−11.06356630052318700046867394716, −10.08927395051771743893684804550, −9.150350739841772685069628447120, −8.696166163424621427674216809504, −7.34146793172562160046139993443, −6.60865865069333408239607336365, −5.66248146017224201443960953748, −4.04063298891995369242558301329, −2.90142231580611624816334042439, −1.13669044249807591949724917468,
1.13669044249807591949724917468, 2.90142231580611624816334042439, 4.04063298891995369242558301329, 5.66248146017224201443960953748, 6.60865865069333408239607336365, 7.34146793172562160046139993443, 8.696166163424621427674216809504, 9.150350739841772685069628447120, 10.08927395051771743893684804550, 11.06356630052318700046867394716
