| L(s) = 1 | − 2-s − 4-s + 3·8-s + 5·13-s − 16-s − 6·17-s + 19-s − 2·23-s − 5·26-s + 31-s − 5·32-s + 6·34-s + 5·37-s − 38-s − 10·41-s − 5·43-s + 2·46-s − 6·47-s − 7·49-s − 5·52-s − 2·53-s − 10·59-s + 61-s − 62-s + 7·64-s − 5·67-s + 6·68-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 1.06·8-s + 1.38·13-s − 1/4·16-s − 1.45·17-s + 0.229·19-s − 0.417·23-s − 0.980·26-s + 0.179·31-s − 0.883·32-s + 1.02·34-s + 0.821·37-s − 0.162·38-s − 1.56·41-s − 0.762·43-s + 0.294·46-s − 0.875·47-s − 49-s − 0.693·52-s − 0.274·53-s − 1.30·59-s + 0.128·61-s − 0.127·62-s + 7/8·64-s − 0.610·67-s + 0.727·68-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8533508835\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8533508835\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | \( 1 \) | |
| 5 | \( 1 \) | |
| 11 | \( 1 \) | |
| good | 2 | \( 1 + T + p T^{2} \) | 1.2.b |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 13 | \( 1 - 5 T + p T^{2} \) | 1.13.af |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - T + p T^{2} \) | 1.31.ab |
| 37 | \( 1 - 5 T + p T^{2} \) | 1.37.af |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 + 5 T + p T^{2} \) | 1.43.f |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 - T + p T^{2} \) | 1.61.ab |
| 67 | \( 1 + 5 T + p T^{2} \) | 1.67.f |
| 71 | \( 1 - 10 T + p T^{2} \) | 1.71.ak |
| 73 | \( 1 - 5 T + p T^{2} \) | 1.73.af |
| 79 | \( 1 - 13 T + p T^{2} \) | 1.79.an |
| 83 | \( 1 - 10 T + p T^{2} \) | 1.83.ak |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 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
−15.33454405623184, −14.82687808406827, −14.01709715377510, −13.47239418082276, −13.42101010305260, −12.68441038079376, −11.97897651590048, −11.22913111945429, −10.94146876771634, −10.38670428332096, −9.591777767510108, −9.360383502795082, −8.611531269601179, −8.197608963349298, −7.857740826603711, −6.721175955684669, −6.602133631182843, −5.704120034250873, −4.946612533915310, −4.426003935464256, −3.752601422256194, −3.101239942033686, −1.964794848936790, −1.426133732463718, −0.4262001719840138,
0.4262001719840138, 1.426133732463718, 1.964794848936790, 3.101239942033686, 3.752601422256194, 4.426003935464256, 4.946612533915310, 5.704120034250873, 6.602133631182843, 6.721175955684669, 7.857740826603711, 8.197608963349298, 8.611531269601179, 9.360383502795082, 9.591777767510108, 10.38670428332096, 10.94146876771634, 11.22913111945429, 11.97897651590048, 12.68441038079376, 13.42101010305260, 13.47239418082276, 14.01709715377510, 14.82687808406827, 15.33454405623184
