| L(s) = 1 | + 3-s − 5-s + 9-s − 11-s − 15-s + 2·17-s − 2·19-s + 4·23-s + 25-s + 27-s − 4·31-s − 33-s + 2·37-s − 12·41-s − 4·43-s − 45-s + 8·47-s + 2·51-s + 10·53-s + 55-s − 2·57-s + 12·59-s + 2·61-s − 4·67-s + 4·69-s − 8·71-s + 75-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1/3·9-s − 0.301·11-s − 0.258·15-s + 0.485·17-s − 0.458·19-s + 0.834·23-s + 1/5·25-s + 0.192·27-s − 0.718·31-s − 0.174·33-s + 0.328·37-s − 1.87·41-s − 0.609·43-s − 0.149·45-s + 1.16·47-s + 0.280·51-s + 1.37·53-s + 0.134·55-s − 0.264·57-s + 1.56·59-s + 0.256·61-s − 0.488·67-s + 0.481·69-s − 0.949·71-s + 0.115·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 258720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 258720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.583286334\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.583286334\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 12 T + p T^{2} \) | 1.41.m |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + p T^{2} \) | 1.73.a |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−13.00750880315879, −12.26849246174666, −11.95549343014218, −11.54487447888778, −10.89282176453286, −10.41187031977033, −10.20285531323137, −9.482657641301791, −9.037606353204655, −8.578740200343617, −8.256798692835139, −7.563267437306100, −7.354407152802376, −6.704558561112043, −6.348781870474513, −5.405831276021351, −5.263445162415591, −4.587569115468837, −3.892913633507841, −3.633373870354612, −2.981307675691483, −2.449395468431816, −1.862233713670780, −1.134765073770095, −0.4449043712054633,
0.4449043712054633, 1.134765073770095, 1.862233713670780, 2.449395468431816, 2.981307675691483, 3.633373870354612, 3.892913633507841, 4.587569115468837, 5.263445162415591, 5.405831276021351, 6.348781870474513, 6.704558561112043, 7.354407152802376, 7.563267437306100, 8.256798692835139, 8.578740200343617, 9.037606353204655, 9.482657641301791, 10.20285531323137, 10.41187031977033, 10.89282176453286, 11.54487447888778, 11.95549343014218, 12.26849246174666, 13.00750880315879
