| L(s) = 1 | − 5-s + 4·7-s − 3·9-s − 11-s − 6·17-s + 4·19-s − 4·23-s + 25-s − 2·29-s + 8·31-s − 4·35-s + 10·37-s − 10·41-s + 3·45-s + 4·47-s + 9·49-s − 10·53-s + 55-s − 4·59-s − 2·61-s − 12·63-s − 8·67-s + 14·73-s − 4·77-s + 16·79-s + 9·81-s − 8·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 1.51·7-s − 9-s − 0.301·11-s − 1.45·17-s + 0.917·19-s − 0.834·23-s + 1/5·25-s − 0.371·29-s + 1.43·31-s − 0.676·35-s + 1.64·37-s − 1.56·41-s + 0.447·45-s + 0.583·47-s + 9/7·49-s − 1.37·53-s + 0.134·55-s − 0.520·59-s − 0.256·61-s − 1.51·63-s − 0.977·67-s + 1.63·73-s − 0.455·77-s + 1.80·79-s + 81-s − 0.878·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 148720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 148720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 11 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 - 16 T + p T^{2} \) | 1.79.aq |
| 83 | \( 1 + 8 T + p T^{2} \) | 1.83.i |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 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
−13.65013581340386, −13.40575320312547, −12.31901032222923, −12.18990556859870, −11.51880944123420, −11.29218326146978, −10.89146438362918, −10.44272365204026, −9.563369670795758, −9.326058155917089, −8.507180673906481, −8.203073136067285, −7.963796324920777, −7.386907374567269, −6.687706003752972, −6.185801171744025, −5.601784677666359, −5.007380426364403, −4.624922561005196, −4.177071844708673, −3.397253362567059, −2.761818923538773, −2.230168941396869, −1.605760336007647, −0.7988554040463604, 0,
0.7988554040463604, 1.605760336007647, 2.230168941396869, 2.761818923538773, 3.397253362567059, 4.177071844708673, 4.624922561005196, 5.007380426364403, 5.601784677666359, 6.185801171744025, 6.687706003752972, 7.386907374567269, 7.963796324920777, 8.203073136067285, 8.507180673906481, 9.326058155917089, 9.563369670795758, 10.44272365204026, 10.89146438362918, 11.29218326146978, 11.51880944123420, 12.18990556859870, 12.31901032222923, 13.40575320312547, 13.65013581340386
