| L(s) = 1 | + 3-s − 4·7-s + 9-s + 2·17-s + 4·19-s − 4·21-s + 8·23-s + 27-s + 2·29-s − 8·31-s + 2·37-s + 6·41-s + 12·43-s + 9·49-s + 2·51-s − 10·53-s + 4·57-s − 10·61-s − 4·63-s + 4·67-s + 8·69-s − 16·71-s − 6·73-s + 8·79-s + 81-s + 4·83-s + 2·87-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.51·7-s + 1/3·9-s + 0.485·17-s + 0.917·19-s − 0.872·21-s + 1.66·23-s + 0.192·27-s + 0.371·29-s − 1.43·31-s + 0.328·37-s + 0.937·41-s + 1.82·43-s + 9/7·49-s + 0.280·51-s − 1.37·53-s + 0.529·57-s − 1.28·61-s − 0.503·63-s + 0.488·67-s + 0.963·69-s − 1.89·71-s − 0.702·73-s + 0.900·79-s + 1/9·81-s + 0.439·83-s + 0.214·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 202800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 202800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 16 T + p T^{2} \) | 1.71.q |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 + 6 T + p T^{2} \) | 1.97.g |
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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
−13.34957613361754, −12.84633655235526, −12.41389605627431, −12.13442591040006, −11.33361859140128, −10.83509711387690, −10.50906597741556, −9.807506812935874, −9.437411212959164, −9.084001873254346, −8.856089727185656, −7.862657963836152, −7.542331798838714, −7.186447113121580, −6.542226139481214, −6.110840352764534, −5.566410866776214, −5.012684181861940, −4.340128293242768, −3.744497712331953, −3.192199976335341, −2.928869096097012, −2.360643222998573, −1.392657201630816, −0.8768880727459601, 0,
0.8768880727459601, 1.392657201630816, 2.360643222998573, 2.928869096097012, 3.192199976335341, 3.744497712331953, 4.340128293242768, 5.012684181861940, 5.566410866776214, 6.110840352764534, 6.542226139481214, 7.186447113121580, 7.542331798838714, 7.862657963836152, 8.856089727185656, 9.084001873254346, 9.437411212959164, 9.807506812935874, 10.50906597741556, 10.83509711387690, 11.33361859140128, 12.13442591040006, 12.41389605627431, 12.84633655235526, 13.34957613361754
