| L(s) = 1 | + 3-s + 2·7-s + 9-s + 2·13-s + 6·17-s − 4·19-s + 2·21-s + 27-s − 2·29-s − 8·31-s + 2·39-s + 2·41-s + 2·43-s − 8·47-s − 3·49-s + 6·51-s + 8·53-s − 4·57-s − 8·59-s − 10·61-s + 2·63-s − 8·67-s − 12·71-s + 14·73-s − 8·79-s + 81-s − 6·83-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.755·7-s + 1/3·9-s + 0.554·13-s + 1.45·17-s − 0.917·19-s + 0.436·21-s + 0.192·27-s − 0.371·29-s − 1.43·31-s + 0.320·39-s + 0.312·41-s + 0.304·43-s − 1.16·47-s − 3/7·49-s + 0.840·51-s + 1.09·53-s − 0.529·57-s − 1.04·59-s − 1.28·61-s + 0.251·63-s − 0.977·67-s − 1.42·71-s + 1.63·73-s − 0.900·79-s + 1/9·81-s − 0.658·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 11 | \( 1 \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + p T^{2} \) | 1.37.a |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 8 T + p T^{2} \) | 1.53.ai |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 97 | \( 1 + 12 T + p T^{2} \) | 1.97.m |
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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
−14.96545733949181, −14.70350977826505, −14.16116049245884, −13.74094160598868, −13.02739677646266, −12.63506286252486, −12.09449309302790, −11.39359267877714, −10.95161303752150, −10.42761509258004, −9.820778104369251, −9.235168207772652, −8.673708635880317, −8.206006438414316, −7.621309989744691, −7.262015063979356, −6.390134143898937, −5.790235228022976, −5.240265653647111, −4.492411547086094, −3.910571274618286, −3.294969877063544, −2.612767807336810, −1.662915260239637, −1.334093870436594, 0,
1.334093870436594, 1.662915260239637, 2.612767807336810, 3.294969877063544, 3.910571274618286, 4.492411547086094, 5.240265653647111, 5.790235228022976, 6.390134143898937, 7.262015063979356, 7.621309989744691, 8.206006438414316, 8.673708635880317, 9.235168207772652, 9.820778104369251, 10.42761509258004, 10.95161303752150, 11.39359267877714, 12.09449309302790, 12.63506286252486, 13.02739677646266, 13.74094160598868, 14.16116049245884, 14.70350977826505, 14.96545733949181
