| L(s) = 1 | + 2-s + 4-s + 5-s − 7-s + 8-s + 10-s − 6·11-s − 14-s + 16-s + 4·19-s + 20-s − 6·22-s + 25-s − 28-s + 4·31-s + 32-s − 35-s + 4·37-s + 4·38-s + 40-s + 6·41-s − 10·43-s − 6·44-s − 6·47-s + 49-s + 50-s − 6·53-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.377·7-s + 0.353·8-s + 0.316·10-s − 1.80·11-s − 0.267·14-s + 1/4·16-s + 0.917·19-s + 0.223·20-s − 1.27·22-s + 1/5·25-s − 0.188·28-s + 0.718·31-s + 0.176·32-s − 0.169·35-s + 0.657·37-s + 0.648·38-s + 0.158·40-s + 0.937·41-s − 1.52·43-s − 0.904·44-s − 0.875·47-s + 1/7·49-s + 0.141·50-s − 0.824·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 106470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 106470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 7 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 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 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 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
−13.79462539565501, −13.45823451584699, −12.88443073002698, −12.79316669400807, −12.06463800387772, −11.50898692961941, −11.09631374250582, −10.43009429623150, −10.13391420037988, −9.651864386900536, −9.088199407063018, −8.322345209597881, −7.823117539265228, −7.520145237765801, −6.760083122762995, −6.287198944630588, −5.750154251615920, −5.220353346444673, −4.845667438700101, −4.266702295998854, −3.286562186501966, −3.063237668706428, −2.441963580333315, −1.812594434892849, −0.9285340510021432, 0,
0.9285340510021432, 1.812594434892849, 2.441963580333315, 3.063237668706428, 3.286562186501966, 4.266702295998854, 4.845667438700101, 5.220353346444673, 5.750154251615920, 6.287198944630588, 6.760083122762995, 7.520145237765801, 7.823117539265228, 8.322345209597881, 9.088199407063018, 9.651864386900536, 10.13391420037988, 10.43009429623150, 11.09631374250582, 11.50898692961941, 12.06463800387772, 12.79316669400807, 12.88443073002698, 13.45823451584699, 13.79462539565501
