| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 8-s + 9-s + 4·11-s − 12-s − 6·13-s + 16-s + 4·17-s + 18-s − 19-s + 4·22-s − 4·23-s − 24-s − 6·26-s − 27-s + 6·29-s + 6·31-s + 32-s − 4·33-s + 4·34-s + 36-s − 10·37-s − 38-s + 6·39-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.353·8-s + 1/3·9-s + 1.20·11-s − 0.288·12-s − 1.66·13-s + 1/4·16-s + 0.970·17-s + 0.235·18-s − 0.229·19-s + 0.852·22-s − 0.834·23-s − 0.204·24-s − 1.17·26-s − 0.192·27-s + 1.11·29-s + 1.07·31-s + 0.176·32-s − 0.696·33-s + 0.685·34-s + 1/6·36-s − 1.64·37-s − 0.162·38-s + 0.960·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 139650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 139650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| 19 | \( 1 + T \) | |
| good | 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + 4 T + p T^{2} \) | 1.41.e |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 - 2 T + p T^{2} \) | 1.83.ac |
| 89 | \( 1 - 8 T + p T^{2} \) | 1.89.ai |
| 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
−13.66736661161958, −13.27960220654655, −12.29487368879095, −12.16664796171510, −11.99219847284517, −11.62716429500383, −10.65015341943870, −10.39573034695282, −9.911772599861025, −9.501231198400800, −8.780614702114294, −8.197335016178457, −7.669066261180282, −7.051857090784233, −6.673147768149152, −6.278415709934802, −5.589919463525711, −5.114338095389035, −4.642425269765814, −4.191995642679015, −3.461702943484848, −3.025878910661092, −2.185432007692730, −1.654741742636988, −0.9014733363542388, 0,
0.9014733363542388, 1.654741742636988, 2.185432007692730, 3.025878910661092, 3.461702943484848, 4.191995642679015, 4.642425269765814, 5.114338095389035, 5.589919463525711, 6.278415709934802, 6.673147768149152, 7.051857090784233, 7.669066261180282, 8.197335016178457, 8.780614702114294, 9.501231198400800, 9.911772599861025, 10.39573034695282, 10.65015341943870, 11.62716429500383, 11.99219847284517, 12.16664796171510, 12.29487368879095, 13.27960220654655, 13.66736661161958
