| L(s) = 1 | − 2·3-s + 5-s − 2·7-s + 9-s + 2·13-s − 2·15-s + 3·17-s + 2·19-s + 4·21-s + 6·23-s − 4·25-s + 4·27-s + 2·29-s + 6·31-s − 2·35-s + 37-s − 4·39-s + 8·43-s + 45-s + 9·47-s − 3·49-s − 6·51-s − 4·57-s − 12·59-s + 8·61-s − 2·63-s + 2·65-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.447·5-s − 0.755·7-s + 1/3·9-s + 0.554·13-s − 0.516·15-s + 0.727·17-s + 0.458·19-s + 0.872·21-s + 1.25·23-s − 4/5·25-s + 0.769·27-s + 0.371·29-s + 1.07·31-s − 0.338·35-s + 0.164·37-s − 0.640·39-s + 1.21·43-s + 0.149·45-s + 1.31·47-s − 3/7·49-s − 0.840·51-s − 0.529·57-s − 1.56·59-s + 1.02·61-s − 0.251·63-s + 0.248·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 71632 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71632 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 37 | \( 1 - T \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 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 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 + 9 T + p T^{2} \) | 1.71.j |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - T + p T^{2} \) | 1.79.ab |
| 83 | \( 1 - 13 T + p T^{2} \) | 1.83.an |
| 89 | \( 1 + 16 T + p T^{2} \) | 1.89.q |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
| 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
−14.30715522592064, −13.65896782540739, −13.46592287438270, −12.71974409922544, −12.33800570672621, −11.89362227301141, −11.35048386013819, −10.87351330694691, −10.36925046219567, −9.957996765106372, −9.325599596548454, −8.962308934944655, −8.197448034966081, −7.593468477671252, −6.993207607821797, −6.373664136868738, −6.084143243604720, −5.534979132927043, −5.101744152042133, −4.386463522979219, −3.725755311490623, −2.953317626651763, −2.553320884013554, −1.326345782806096, −0.9340903666192820, 0,
0.9340903666192820, 1.326345782806096, 2.553320884013554, 2.953317626651763, 3.725755311490623, 4.386463522979219, 5.101744152042133, 5.534979132927043, 6.084143243604720, 6.373664136868738, 6.993207607821797, 7.593468477671252, 8.197448034966081, 8.962308934944655, 9.325599596548454, 9.957996765106372, 10.36925046219567, 10.87351330694691, 11.35048386013819, 11.89362227301141, 12.33800570672621, 12.71974409922544, 13.46592287438270, 13.65896782540739, 14.30715522592064
