| L(s) = 1 | − 3·5-s − 4·13-s − 6·17-s + 4·19-s + 6·23-s + 4·25-s + 3·29-s − 8·31-s + 8·37-s + 6·41-s − 8·43-s + 6·47-s − 9·53-s − 3·59-s − 10·61-s + 12·65-s + 10·67-s + 6·71-s − 7·73-s − 17·79-s − 12·83-s + 18·85-s − 6·89-s − 12·95-s − 10·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 1.34·5-s − 1.10·13-s − 1.45·17-s + 0.917·19-s + 1.25·23-s + 4/5·25-s + 0.557·29-s − 1.43·31-s + 1.31·37-s + 0.937·41-s − 1.21·43-s + 0.875·47-s − 1.23·53-s − 0.390·59-s − 1.28·61-s + 1.48·65-s + 1.22·67-s + 0.712·71-s − 0.819·73-s − 1.91·79-s − 1.31·83-s + 1.95·85-s − 0.635·89-s − 1.23·95-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7312667568\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7312667568\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| good | 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 10 T + p T^{2} \) | 1.67.ak |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 + 7 T + p T^{2} \) | 1.73.h |
| 79 | \( 1 + 17 T + p T^{2} \) | 1.79.r |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−15.58477430772963, −15.17694342751004, −14.54466230365784, −14.14177691741431, −13.25668717991979, −12.80105024602393, −12.34677060052290, −11.62635756247026, −11.25450001526624, −10.88503171538877, −10.03370833272390, −9.333464221487135, −8.964358994705181, −8.187813775568916, −7.629437640223279, −7.167899910994504, −6.710720294483216, −5.773531604619743, −4.983558425897907, −4.504659941594588, −3.939242070631936, −3.055020573585702, −2.578455332064194, −1.444270068471389, −0.3555638399950051,
0.3555638399950051, 1.444270068471389, 2.578455332064194, 3.055020573585702, 3.939242070631936, 4.504659941594588, 4.983558425897907, 5.773531604619743, 6.710720294483216, 7.167899910994504, 7.629437640223279, 8.187813775568916, 8.964358994705181, 9.333464221487135, 10.03370833272390, 10.88503171538877, 11.25450001526624, 11.62635756247026, 12.34677060052290, 12.80105024602393, 13.25668717991979, 14.14177691741431, 14.54466230365784, 15.17694342751004, 15.58477430772963
