| L(s) = 1 | + 2·3-s + 5-s + 2·7-s + 9-s + 13-s + 2·15-s + 3·17-s + 2·19-s + 4·21-s − 6·23-s − 4·25-s − 4·27-s + 29-s + 10·31-s + 2·35-s − 3·37-s + 2·39-s + 11·41-s + 12·43-s + 45-s + 10·47-s − 3·49-s + 6·51-s + 9·53-s + 4·57-s − 4·59-s + 6·61-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.447·5-s + 0.755·7-s + 1/3·9-s + 0.277·13-s + 0.516·15-s + 0.727·17-s + 0.458·19-s + 0.872·21-s − 1.25·23-s − 4/5·25-s − 0.769·27-s + 0.185·29-s + 1.79·31-s + 0.338·35-s − 0.493·37-s + 0.320·39-s + 1.71·41-s + 1.82·43-s + 0.149·45-s + 1.45·47-s − 3/7·49-s + 0.840·51-s + 1.23·53-s + 0.529·57-s − 0.520·59-s + 0.768·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3872 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.604666410\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.604666410\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - T + p T^{2} \) | 1.29.ab |
| 31 | \( 1 - 10 T + p T^{2} \) | 1.31.ak |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 - 11 T + p T^{2} \) | 1.41.al |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 - 10 T + p T^{2} \) | 1.47.ak |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 2 T + p T^{2} \) | 1.79.c |
| 83 | \( 1 + 14 T + p T^{2} \) | 1.83.o |
| 89 | \( 1 + T + p T^{2} \) | 1.89.b |
| 97 | \( 1 - 11 T + p T^{2} \) | 1.97.al |
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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
−8.535240266317922487385019685679, −7.71990293041314627611970698480, −7.45745214242486139816618070133, −6.03071280739939111680869189764, −5.70313108829488008977187510841, −4.48091665258988671468461695129, −3.84427585872851509526933245739, −2.79565924032788372310580670424, −2.17482249812905007594416584983, −1.11200571704776262206779926037,
1.11200571704776262206779926037, 2.17482249812905007594416584983, 2.79565924032788372310580670424, 3.84427585872851509526933245739, 4.48091665258988671468461695129, 5.70313108829488008977187510841, 6.03071280739939111680869189764, 7.45745214242486139816618070133, 7.71990293041314627611970698480, 8.535240266317922487385019685679
