| L(s) = 1 | − 3-s + 5-s + 2·7-s + 9-s − 2·11-s + 2·13-s − 15-s − 6·19-s − 2·21-s − 6·23-s + 25-s − 27-s + 6·29-s − 8·31-s + 2·33-s + 2·35-s + 6·37-s − 2·39-s − 6·41-s − 43-s + 45-s + 2·47-s − 3·49-s + 14·53-s − 2·55-s + 6·57-s − 6·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.755·7-s + 1/3·9-s − 0.603·11-s + 0.554·13-s − 0.258·15-s − 1.37·19-s − 0.436·21-s − 1.25·23-s + 1/5·25-s − 0.192·27-s + 1.11·29-s − 1.43·31-s + 0.348·33-s + 0.338·35-s + 0.986·37-s − 0.320·39-s − 0.937·41-s − 0.152·43-s + 0.149·45-s + 0.291·47-s − 3/7·49-s + 1.92·53-s − 0.269·55-s + 0.794·57-s − 0.781·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 41280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 - T \) | |
| 43 | \( 1 + T \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 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.g |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 - 14 T + p T^{2} \) | 1.53.ao |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 8 T + p T^{2} \) | 1.73.ai |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 8 T + p T^{2} \) | 1.83.ai |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−14.84646525348688, −14.63162912330587, −13.92643523643870, −13.35756076866551, −13.03143127196114, −12.33381962821027, −11.92375424168589, −11.27569368365446, −10.78861656363749, −10.41003055357937, −9.913834636080144, −9.203883800759019, −8.439545457651807, −8.256274407587283, −7.498743963847197, −6.848785923824194, −6.242148426786040, −5.768487335168689, −5.231815554042437, −4.539179322405898, −4.092940830028463, −3.267811982793561, −2.239949492019167, −1.912294690975184, −0.9745390444972573, 0,
0.9745390444972573, 1.912294690975184, 2.239949492019167, 3.267811982793561, 4.092940830028463, 4.539179322405898, 5.231815554042437, 5.768487335168689, 6.242148426786040, 6.848785923824194, 7.498743963847197, 8.256274407587283, 8.439545457651807, 9.203883800759019, 9.913834636080144, 10.41003055357937, 10.78861656363749, 11.27569368365446, 11.92375424168589, 12.33381962821027, 13.03143127196114, 13.35756076866551, 13.92643523643870, 14.63162912330587, 14.84646525348688
