| L(s) = 1 | − 2·2-s + 2·4-s − 7-s + 2·11-s − 13-s + 2·14-s − 4·16-s − 4·17-s + 3·19-s − 4·22-s − 9·23-s + 2·26-s − 2·28-s + 29-s − 5·31-s + 8·32-s + 8·34-s + 8·37-s − 6·38-s − 6·41-s + 9·43-s + 4·44-s + 18·46-s − 3·47-s + 49-s − 2·52-s + 3·53-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s − 0.377·7-s + 0.603·11-s − 0.277·13-s + 0.534·14-s − 16-s − 0.970·17-s + 0.688·19-s − 0.852·22-s − 1.87·23-s + 0.392·26-s − 0.377·28-s + 0.185·29-s − 0.898·31-s + 1.41·32-s + 1.37·34-s + 1.31·37-s − 0.973·38-s − 0.937·41-s + 1.37·43-s + 0.603·44-s + 2.65·46-s − 0.437·47-s + 1/7·49-s − 0.277·52-s + 0.412·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5389144276\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5389144276\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 13 | \( 1 + T \) | |
| good | 2 | \( 1 + p T + p T^{2} \) | 1.2.c |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 - 3 T + p T^{2} \) | 1.19.ad |
| 23 | \( 1 + 9 T + p T^{2} \) | 1.23.j |
| 29 | \( 1 - T + p T^{2} \) | 1.29.ab |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 9 T + p T^{2} \) | 1.43.aj |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 5 T + p T^{2} \) | 1.73.f |
| 79 | \( 1 + 13 T + p T^{2} \) | 1.79.n |
| 83 | \( 1 + 11 T + p T^{2} \) | 1.83.l |
| 89 | \( 1 + T + p T^{2} \) | 1.89.b |
| 97 | \( 1 + T + p T^{2} \) | 1.97.b |
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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
−15.94204484315187, −15.34789068769038, −14.47931960070556, −14.14543865327964, −13.39620872068891, −12.92744514236004, −12.17152185366500, −11.45958347714276, −11.31449960513569, −10.33789330511288, −9.995493261296479, −9.545742519267961, −8.827407364987852, −8.599096140415405, −7.640151991494499, −7.431186452390231, −6.611388241412069, −6.129452906777726, −5.333810213464571, −4.322278947798326, −3.960778091570047, −2.836335878487805, −2.113864442681334, −1.390806693958604, −0.3927121650580538,
0.3927121650580538, 1.390806693958604, 2.113864442681334, 2.836335878487805, 3.960778091570047, 4.322278947798326, 5.333810213464571, 6.129452906777726, 6.611388241412069, 7.431186452390231, 7.640151991494499, 8.599096140415405, 8.827407364987852, 9.545742519267961, 9.995493261296479, 10.33789330511288, 11.31449960513569, 11.45958347714276, 12.17152185366500, 12.92744514236004, 13.39620872068891, 14.14543865327964, 14.47931960070556, 15.34789068769038, 15.94204484315187
