| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 8-s + 9-s − 12-s − 5·13-s + 16-s + 6·17-s − 18-s + 7·19-s − 6·23-s + 24-s + 5·26-s − 27-s − 8·31-s − 32-s − 6·34-s + 36-s − 37-s − 7·38-s + 5·39-s + 8·43-s + 6·46-s − 6·47-s − 48-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 − 0.288·12-s − 1.38·13-s + 1/4·16-s + 1.45·17-s − 0.235·18-s + 1.60·19-s − 1.25·23-s + 0.204·24-s + 0.980·26-s − 0.192·27-s − 1.43·31-s − 0.176·32-s − 1.02·34-s + 1/6·36-s − 0.164·37-s − 1.13·38-s + 0.800·39-s + 1.21·43-s + 0.884·46-s − 0.875·47-s − 0.144·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| good | 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 5 T + p T^{2} \) | 1.13.f |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 7 T + p T^{2} \) | 1.19.ah |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + T + p T^{2} \) | 1.37.b |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 - T + p T^{2} \) | 1.61.ab |
| 67 | \( 1 + 13 T + p T^{2} \) | 1.67.n |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 5 T + p T^{2} \) | 1.73.f |
| 79 | \( 1 + 7 T + p T^{2} \) | 1.79.h |
| 83 | \( 1 - 18 T + p T^{2} \) | 1.83.as |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 7 T + p T^{2} \) | 1.97.ah |
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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
−7.65109861048481194585512537700, −7.09425148965284136036381602484, −6.15997819673331064505933601889, −5.46746767928956906435806540590, −5.00039473757932147449809338334, −3.85001089347771845206232372011, −3.06155084127255832852186929223, −2.05666066872417321305878249951, −1.10488875166942823009191099151, 0,
1.10488875166942823009191099151, 2.05666066872417321305878249951, 3.06155084127255832852186929223, 3.85001089347771845206232372011, 5.00039473757932147449809338334, 5.46746767928956906435806540590, 6.15997819673331064505933601889, 7.09425148965284136036381602484, 7.65109861048481194585512537700
