| L(s) = 1 | + 3-s − 2·7-s + 9-s − 4·11-s + 6·13-s − 4·17-s − 2·19-s − 2·21-s − 5·25-s + 27-s − 2·29-s − 4·31-s − 4·33-s + 2·37-s + 6·39-s + 2·41-s − 10·43-s − 3·49-s − 4·51-s − 12·53-s − 2·57-s − 12·59-s − 6·61-s − 2·63-s + 10·67-s − 8·71-s − 14·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.755·7-s + 1/3·9-s − 1.20·11-s + 1.66·13-s − 0.970·17-s − 0.458·19-s − 0.436·21-s − 25-s + 0.192·27-s − 0.371·29-s − 0.718·31-s − 0.696·33-s + 0.328·37-s + 0.960·39-s + 0.312·41-s − 1.52·43-s − 3/7·49-s − 0.560·51-s − 1.64·53-s − 0.264·57-s − 1.56·59-s − 0.768·61-s − 0.251·63-s + 1.22·67-s − 0.949·71-s − 1.63·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 101568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 101568 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7094505497\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7094505497\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 23 | \( 1 \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 - 10 T + p T^{2} \) | 1.67.ak |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 16 T + p T^{2} \) | 1.89.aq |
| 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
−13.56665915065462, −13.14505903560424, −13.10011019878189, −12.48194449756724, −11.72985641996548, −11.18808345507802, −10.72443605460741, −10.41515768477572, −9.651461239484754, −9.304841438785193, −8.785372013069369, −8.195069124925486, −7.902569638272512, −7.273658715704551, −6.545532198668732, −6.216889311632087, −5.706562539556510, −4.951399328279803, −4.351282889741778, −3.743829868166645, −3.236631562469688, −2.750774279784863, −1.920071555549694, −1.485444283788638, −0.2389043366507000,
0.2389043366507000, 1.485444283788638, 1.920071555549694, 2.750774279784863, 3.236631562469688, 3.743829868166645, 4.351282889741778, 4.951399328279803, 5.706562539556510, 6.216889311632087, 6.545532198668732, 7.273658715704551, 7.902569638272512, 8.195069124925486, 8.785372013069369, 9.304841438785193, 9.651461239484754, 10.41515768477572, 10.72443605460741, 11.18808345507802, 11.72985641996548, 12.48194449756724, 13.10011019878189, 13.14505903560424, 13.56665915065462
