| L(s) = 1 | − 3-s − 2·5-s − 7-s + 9-s − 6·13-s + 2·15-s − 2·17-s + 4·19-s + 21-s + 4·23-s − 25-s − 27-s + 10·29-s + 8·31-s + 2·35-s − 6·37-s + 6·39-s − 2·41-s − 4·43-s − 2·45-s − 8·47-s + 49-s + 2·51-s + 10·53-s − 4·57-s + 12·59-s + 2·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s − 0.377·7-s + 1/3·9-s − 1.66·13-s + 0.516·15-s − 0.485·17-s + 0.917·19-s + 0.218·21-s + 0.834·23-s − 1/5·25-s − 0.192·27-s + 1.85·29-s + 1.43·31-s + 0.338·35-s − 0.986·37-s + 0.960·39-s − 0.312·41-s − 0.609·43-s − 0.298·45-s − 1.16·47-s + 1/7·49-s + 0.280·51-s + 1.37·53-s − 0.529·57-s + 1.56·59-s + 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8472770573\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8472770573\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 7 | \( 1 + T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−9.907669061665662410086615419667, −8.742756249780573873818038490305, −7.909598986797872682521861264692, −7.04895830473267504072201598594, −6.55534782210605698363840663093, −5.17124547767097325451854702106, −4.71122082833635473974871484179, −3.54742556842302716727388578393, −2.49703074453830152772273364467, −0.67099960202093059418976869535,
0.67099960202093059418976869535, 2.49703074453830152772273364467, 3.54742556842302716727388578393, 4.71122082833635473974871484179, 5.17124547767097325451854702106, 6.55534782210605698363840663093, 7.04895830473267504072201598594, 7.909598986797872682521861264692, 8.742756249780573873818038490305, 9.907669061665662410086615419667
