| L(s) = 1 | + 2·5-s + 2·7-s − 6·11-s + 13-s − 6·17-s + 4·19-s − 25-s + 9·29-s + 3·31-s + 4·35-s + 2·37-s + 41-s + 47-s − 3·49-s − 12·53-s − 12·55-s − 12·59-s − 8·61-s + 2·65-s + 10·67-s + 13·71-s + 5·73-s − 12·77-s − 4·79-s + 2·83-s − 12·85-s + 12·89-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 0.755·7-s − 1.80·11-s + 0.277·13-s − 1.45·17-s + 0.917·19-s − 1/5·25-s + 1.67·29-s + 0.538·31-s + 0.676·35-s + 0.328·37-s + 0.156·41-s + 0.145·47-s − 3/7·49-s − 1.64·53-s − 1.61·55-s − 1.56·59-s − 1.02·61-s + 0.248·65-s + 1.22·67-s + 1.54·71-s + 0.585·73-s − 1.36·77-s − 0.450·79-s + 0.219·83-s − 1.30·85-s + 1.27·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 23 | \( 1 \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 29 | \( 1 - 9 T + p T^{2} \) | 1.29.aj |
| 31 | \( 1 - 3 T + p T^{2} \) | 1.31.ad |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - T + p T^{2} \) | 1.41.ab |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 - T + p T^{2} \) | 1.47.ab |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 - 10 T + p T^{2} \) | 1.67.ak |
| 71 | \( 1 - 13 T + p T^{2} \) | 1.71.an |
| 73 | \( 1 - 5 T + p T^{2} \) | 1.73.af |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 - 2 T + p T^{2} \) | 1.83.ac |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 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
−12.95115088686360, −12.51882997634043, −12.04829857308559, −11.35488602495244, −11.00178792209590, −10.71867945128114, −10.14090900270519, −9.737878631705030, −9.321987701395000, −8.690775083626858, −8.222760024043851, −7.879017967524473, −7.455468635249849, −6.749611523303359, −6.169834144344604, −6.004095268101984, −5.097715061237394, −4.909618669635492, −4.623234222397047, −3.736207932685826, −3.004679841213276, −2.587761648196288, −2.115910871979575, −1.541296997918610, −0.8153135470691976, 0,
0.8153135470691976, 1.541296997918610, 2.115910871979575, 2.587761648196288, 3.004679841213276, 3.736207932685826, 4.623234222397047, 4.909618669635492, 5.097715061237394, 6.004095268101984, 6.169834144344604, 6.749611523303359, 7.455468635249849, 7.879017967524473, 8.222760024043851, 8.690775083626858, 9.321987701395000, 9.737878631705030, 10.14090900270519, 10.71867945128114, 11.00178792209590, 11.35488602495244, 12.04829857308559, 12.51882997634043, 12.95115088686360
