| L(s) = 1 | − 3-s − 5-s − 2·7-s − 2·9-s + 11-s + 15-s − 4·17-s − 2·19-s + 2·21-s − 7·23-s − 4·25-s + 5·27-s + 2·29-s − 3·31-s − 33-s + 2·35-s − 11·37-s − 10·41-s − 4·43-s + 2·45-s − 4·47-s − 3·49-s + 4·51-s − 2·53-s − 55-s + 2·57-s + 59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.755·7-s − 2/3·9-s + 0.301·11-s + 0.258·15-s − 0.970·17-s − 0.458·19-s + 0.436·21-s − 1.45·23-s − 4/5·25-s + 0.962·27-s + 0.371·29-s − 0.538·31-s − 0.174·33-s + 0.338·35-s − 1.80·37-s − 1.56·41-s − 0.609·43-s + 0.298·45-s − 0.583·47-s − 3/7·49-s + 0.560·51-s − 0.274·53-s − 0.134·55-s + 0.264·57-s + 0.130·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 118976 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 118976 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + 7 T + p T^{2} \) | 1.23.h |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 3 T + p T^{2} \) | 1.31.d |
| 37 | \( 1 + 11 T + p T^{2} \) | 1.37.l |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 - T + p T^{2} \) | 1.59.ab |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - T + p T^{2} \) | 1.67.ab |
| 71 | \( 1 + 9 T + p T^{2} \) | 1.71.j |
| 73 | \( 1 - 16 T + p T^{2} \) | 1.73.aq |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 7 T + p T^{2} \) | 1.89.ah |
| 97 | \( 1 - 13 T + p T^{2} \) | 1.97.an |
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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
−13.68521070497503, −13.43225378965053, −12.75990415655052, −12.17543698686650, −11.94491722916752, −11.49500862338815, −10.93439824724352, −10.49237653556566, −9.945930519758913, −9.512557583704592, −8.838297781424240, −8.312869693712866, −8.138243383289605, −7.141567550124389, −6.790570221833842, −6.321494250200117, −5.861152171333978, −5.268967653364529, −4.681270011824739, −4.082379886117379, −3.454917107809313, −3.117190210912822, −2.076633477908571, −1.762696947440945, −0.4564353958715830, 0,
0.4564353958715830, 1.762696947440945, 2.076633477908571, 3.117190210912822, 3.454917107809313, 4.082379886117379, 4.681270011824739, 5.268967653364529, 5.861152171333978, 6.321494250200117, 6.790570221833842, 7.141567550124389, 8.138243383289605, 8.312869693712866, 8.838297781424240, 9.512557583704592, 9.945930519758913, 10.49237653556566, 10.93439824724352, 11.49500862338815, 11.94491722916752, 12.17543698686650, 12.75990415655052, 13.43225378965053, 13.68521070497503
