| L(s) = 1 | + 3-s − 5-s − 2·7-s + 9-s − 2·13-s − 15-s + 2·17-s − 8·19-s − 2·21-s + 6·23-s + 25-s + 27-s + 2·29-s + 2·35-s − 4·37-s − 2·39-s − 2·41-s + 43-s − 45-s + 6·47-s − 3·49-s + 2·51-s + 4·53-s − 8·57-s + 8·61-s − 2·63-s + 2·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.755·7-s + 1/3·9-s − 0.554·13-s − 0.258·15-s + 0.485·17-s − 1.83·19-s − 0.436·21-s + 1.25·23-s + 1/5·25-s + 0.192·27-s + 0.371·29-s + 0.338·35-s − 0.657·37-s − 0.320·39-s − 0.312·41-s + 0.152·43-s − 0.149·45-s + 0.875·47-s − 3/7·49-s + 0.280·51-s + 0.549·53-s − 1.05·57-s + 1.02·61-s − 0.251·63-s + 0.248·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 41280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 - T \) | |
| 5 | \( 1 + T \) | |
| 43 | \( 1 - T \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 + 16 T + p T^{2} \) | 1.83.q |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
| 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
−15.01117319101542, −14.58284847350165, −14.05662894811048, −13.28091058996772, −13.02836222670681, −12.42311963950506, −12.09619677342334, −11.34824969844120, −10.63058163798631, −10.38511330236284, −9.675331135338826, −9.153090478474010, −8.611254509047067, −8.211326755534911, −7.452043307643222, −6.971685425697625, −6.520047168210940, −5.814565415576473, −5.020162461759440, −4.463213608176507, −3.803471623060922, −3.231353681396734, −2.610954448561291, −1.962420528843458, −0.9156066593716986, 0,
0.9156066593716986, 1.962420528843458, 2.610954448561291, 3.231353681396734, 3.803471623060922, 4.463213608176507, 5.020162461759440, 5.814565415576473, 6.520047168210940, 6.971685425697625, 7.452043307643222, 8.211326755534911, 8.611254509047067, 9.153090478474010, 9.675331135338826, 10.38511330236284, 10.63058163798631, 11.34824969844120, 12.09619677342334, 12.42311963950506, 13.02836222670681, 13.28091058996772, 14.05662894811048, 14.58284847350165, 15.01117319101542
