| L(s) = 1 | + 2-s − 4-s + 7-s − 3·8-s − 2·13-s + 14-s − 16-s − 3·17-s − 3·19-s + 23-s − 2·26-s − 28-s − 6·29-s + 2·31-s + 5·32-s − 3·34-s + 3·37-s − 3·38-s − 3·41-s − 12·43-s + 46-s − 47-s − 6·49-s + 2·52-s − 6·53-s − 3·56-s − 6·58-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s + 0.377·7-s − 1.06·8-s − 0.554·13-s + 0.267·14-s − 1/4·16-s − 0.727·17-s − 0.688·19-s + 0.208·23-s − 0.392·26-s − 0.188·28-s − 1.11·29-s + 0.359·31-s + 0.883·32-s − 0.514·34-s + 0.493·37-s − 0.486·38-s − 0.468·41-s − 1.82·43-s + 0.147·46-s − 0.145·47-s − 6/7·49-s + 0.277·52-s − 0.824·53-s − 0.400·56-s − 0.787·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.294128448\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.294128448\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | \( 1 \) | |
| 5 | \( 1 \) | |
| 11 | \( 1 \) | |
| good | 2 | \( 1 - T + p T^{2} \) | 1.2.ab |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 + 3 T + p T^{2} \) | 1.19.d |
| 23 | \( 1 - T + p T^{2} \) | 1.23.ab |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 - 3 T + p T^{2} \) | 1.37.ad |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 + T + p T^{2} \) | 1.47.b |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 3 T + p T^{2} \) | 1.59.ad |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 6 T + p T^{2} \) | 1.67.ag |
| 71 | \( 1 - 7 T + p T^{2} \) | 1.71.ah |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 7 T + p T^{2} \) | 1.79.ah |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 + 3 T + p T^{2} \) | 1.97.d |
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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
−15.02609360723604, −14.82541361338529, −14.14653634879681, −13.66410697760436, −13.09370958703501, −12.77647699500784, −12.16513527828036, −11.51691549524766, −11.16671474978618, −10.37942205532277, −9.765736284290644, −9.277666156920061, −8.661873514081188, −8.160850574611506, −7.546244446787937, −6.670987729137717, −6.302651845892963, −5.515110679958429, −4.829027393226307, −4.631243170302281, −3.768412030305528, −3.227167142363433, −2.363208773419578, −1.645967378560796, −0.3856026146315220,
0.3856026146315220, 1.645967378560796, 2.363208773419578, 3.227167142363433, 3.768412030305528, 4.631243170302281, 4.829027393226307, 5.515110679958429, 6.302651845892963, 6.670987729137717, 7.546244446787937, 8.160850574611506, 8.661873514081188, 9.277666156920061, 9.765736284290644, 10.37942205532277, 11.16671474978618, 11.51691549524766, 12.16513527828036, 12.77647699500784, 13.09370958703501, 13.66410697760436, 14.14653634879681, 14.82541361338529, 15.02609360723604
