| L(s) = 1 | + 3-s + 5-s − 7-s − 2·9-s + 15-s + 17-s + 4·19-s − 21-s + 8·23-s − 4·25-s − 5·27-s + 8·29-s − 35-s − 7·37-s − 8·41-s + 11·43-s − 2·45-s − 47-s − 6·49-s + 51-s + 2·53-s + 4·57-s + 14·59-s + 8·61-s + 2·63-s + 8·67-s + 8·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.377·7-s − 2/3·9-s + 0.258·15-s + 0.242·17-s + 0.917·19-s − 0.218·21-s + 1.66·23-s − 4/5·25-s − 0.962·27-s + 1.48·29-s − 0.169·35-s − 1.15·37-s − 1.24·41-s + 1.67·43-s − 0.298·45-s − 0.145·47-s − 6/7·49-s + 0.140·51-s + 0.274·53-s + 0.529·57-s + 1.82·59-s + 1.02·61-s + 0.251·63-s + 0.977·67-s + 0.963·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 327184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 327184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 17 | \( 1 - T + p T^{2} \) | 1.17.ab |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 43 | \( 1 - 11 T + p T^{2} \) | 1.43.al |
| 47 | \( 1 + T + p T^{2} \) | 1.47.b |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 14 T + p T^{2} \) | 1.59.ao |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 9 T + p T^{2} \) | 1.71.aj |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 - 2 T + p T^{2} \) | 1.79.ac |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 4 T + p T^{2} \) | 1.89.ae |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.05694585951625, −12.33739123788061, −11.93365222751468, −11.53682930084855, −10.99822776910307, −10.52192417047106, −10.00742081040963, −9.552052521503987, −9.262089402303831, −8.691134731286182, −8.295505985208081, −7.900149199049353, −7.235270620473451, −6.730217601142273, −6.488565818056741, −5.647380833312609, −5.284169040170967, −5.049691075300398, −4.098851092560422, −3.625102544865610, −3.150817422539720, −2.615297386325231, −2.274603976617495, −1.370265057699800, −0.8897830921769184, 0,
0.8897830921769184, 1.370265057699800, 2.274603976617495, 2.615297386325231, 3.150817422539720, 3.625102544865610, 4.098851092560422, 5.049691075300398, 5.284169040170967, 5.647380833312609, 6.488565818056741, 6.730217601142273, 7.235270620473451, 7.900149199049353, 8.295505985208081, 8.691134731286182, 9.262089402303831, 9.552052521503987, 10.00742081040963, 10.52192417047106, 10.99822776910307, 11.53682930084855, 11.93365222751468, 12.33739123788061, 13.05694585951625
