| L(s) = 1 | + 2-s + 4-s + 8-s + 6·11-s + 13-s + 16-s − 6·19-s + 6·22-s + 6·23-s + 26-s − 2·29-s − 4·31-s + 32-s − 10·37-s − 6·38-s − 6·41-s − 8·43-s + 6·44-s + 6·46-s − 8·47-s + 52-s − 6·53-s − 2·58-s + 10·59-s + 6·61-s − 4·62-s + 64-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s + 1.80·11-s + 0.277·13-s + 1/4·16-s − 1.37·19-s + 1.27·22-s + 1.25·23-s + 0.196·26-s − 0.371·29-s − 0.718·31-s + 0.176·32-s − 1.64·37-s − 0.973·38-s − 0.937·41-s − 1.21·43-s + 0.904·44-s + 0.884·46-s − 1.16·47-s + 0.138·52-s − 0.824·53-s − 0.262·58-s + 1.30·59-s + 0.768·61-s − 0.508·62-s + 1/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 286650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 286650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.238585375\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.238585375\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 10 T + p T^{2} \) | 1.59.ak |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 16 T + p T^{2} \) | 1.79.aq |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−12.72391796529751, −12.28861243726792, −11.90069238879412, −11.35655482360708, −11.00648568294543, −10.68554581539732, −9.844699368232491, −9.648499388273689, −8.918828132843723, −8.536021700078591, −8.280414536243214, −7.348331132206486, −6.871240787781435, −6.631206375387766, −6.260707335154755, −5.539922115285467, −5.021745323256448, −4.651113290860570, −3.886662188726804, −3.616290513714777, −3.233388904673973, −2.320638885734777, −1.729357786123443, −1.400804370488754, −0.4684893266228026,
0.4684893266228026, 1.400804370488754, 1.729357786123443, 2.320638885734777, 3.233388904673973, 3.616290513714777, 3.886662188726804, 4.651113290860570, 5.021745323256448, 5.539922115285467, 6.260707335154755, 6.631206375387766, 6.871240787781435, 7.348331132206486, 8.280414536243214, 8.536021700078591, 8.918828132843723, 9.648499388273689, 9.844699368232491, 10.68554581539732, 11.00648568294543, 11.35655482360708, 11.90069238879412, 12.28861243726792, 12.72391796529751
