| L(s) = 1 | − 5-s − 7-s + 13-s − 6·17-s + 4·19-s + 25-s − 10·29-s + 8·31-s + 35-s − 10·37-s + 6·41-s + 4·43-s + 4·47-s + 49-s − 6·53-s − 2·61-s − 65-s − 4·67-s + 8·71-s + 6·73-s + 8·79-s − 4·83-s + 6·85-s + 6·89-s − 91-s − 4·95-s + 14·97-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.377·7-s + 0.277·13-s − 1.45·17-s + 0.917·19-s + 1/5·25-s − 1.85·29-s + 1.43·31-s + 0.169·35-s − 1.64·37-s + 0.937·41-s + 0.609·43-s + 0.583·47-s + 1/7·49-s − 0.824·53-s − 0.256·61-s − 0.124·65-s − 0.488·67-s + 0.949·71-s + 0.702·73-s + 0.900·79-s − 0.439·83-s + 0.650·85-s + 0.635·89-s − 0.104·91-s − 0.410·95-s + 1.42·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 131040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 131040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 7 | \( 1 + T \) | |
| 13 | \( 1 - T \) | |
| good | 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 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 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−13.68108626436724, −13.34427867712202, −12.65722247407599, −12.35739010906306, −11.78116986608613, −11.21969412048924, −10.97345585709405, −10.40191846430061, −9.807018156175116, −9.187432229813661, −9.010964558780659, −8.367594109089200, −7.685405013813376, −7.451083998952823, −6.676299506558938, −6.428421782855614, −5.703851954019200, −5.173458551510613, −4.601073452463018, −3.949936441452143, −3.593229749499827, −2.870507507549017, −2.292750256791690, −1.583495280779497, −0.7421524648483159, 0,
0.7421524648483159, 1.583495280779497, 2.292750256791690, 2.870507507549017, 3.593229749499827, 3.949936441452143, 4.601073452463018, 5.173458551510613, 5.703851954019200, 6.428421782855614, 6.676299506558938, 7.451083998952823, 7.685405013813376, 8.367594109089200, 9.010964558780659, 9.187432229813661, 9.807018156175116, 10.40191846430061, 10.97345585709405, 11.21969412048924, 11.78116986608613, 12.35739010906306, 12.65722247407599, 13.34427867712202, 13.68108626436724
