| L(s) = 1 | + 3-s + 3·5-s − 4·7-s + 9-s + 3·15-s + 6·17-s + 5·19-s − 4·21-s + 23-s + 4·25-s + 27-s + 8·29-s − 10·31-s − 12·35-s − 4·37-s + 10·41-s + 9·43-s + 3·45-s − 11·47-s + 9·49-s + 6·51-s − 9·53-s + 5·57-s + 5·59-s − 4·63-s − 5·67-s + 69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.34·5-s − 1.51·7-s + 1/3·9-s + 0.774·15-s + 1.45·17-s + 1.14·19-s − 0.872·21-s + 0.208·23-s + 4/5·25-s + 0.192·27-s + 1.48·29-s − 1.79·31-s − 2.02·35-s − 0.657·37-s + 1.56·41-s + 1.37·43-s + 0.447·45-s − 1.60·47-s + 9/7·49-s + 0.840·51-s − 1.23·53-s + 0.662·57-s + 0.650·59-s − 0.503·63-s − 0.610·67-s + 0.120·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 186576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 186576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.515473929\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.515473929\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| 23 | \( 1 - T \) | |
| good | 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 - 9 T + p T^{2} \) | 1.43.aj |
| 47 | \( 1 + 11 T + p T^{2} \) | 1.47.l |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 - 5 T + p T^{2} \) | 1.59.af |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 + 5 T + p T^{2} \) | 1.67.f |
| 71 | \( 1 - 15 T + p T^{2} \) | 1.71.ap |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 9 T + p T^{2} \) | 1.79.j |
| 83 | \( 1 - 2 T + p T^{2} \) | 1.83.ac |
| 89 | \( 1 - 3 T + p T^{2} \) | 1.89.ad |
| 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
−12.96634844280359, −12.78349495363762, −12.51406531544725, −11.81670257138706, −11.20680268008141, −10.45235929262318, −10.22173646197283, −9.693329468065959, −9.321721919333816, −9.232681010463234, −8.436412755264581, −7.763865386508935, −7.382639406095249, −6.780614192555422, −6.320107608170314, −5.823942834234467, −5.436424669736273, −4.898325715506123, −4.045137719322645, −3.324654581859503, −3.174938559494376, −2.563560463517653, −1.905011590734938, −1.220574062384814, −0.6075847161508503,
0.6075847161508503, 1.220574062384814, 1.905011590734938, 2.563560463517653, 3.174938559494376, 3.324654581859503, 4.045137719322645, 4.898325715506123, 5.436424669736273, 5.823942834234467, 6.320107608170314, 6.780614192555422, 7.382639406095249, 7.763865386508935, 8.436412755264581, 9.232681010463234, 9.321721919333816, 9.693329468065959, 10.22173646197283, 10.45235929262318, 11.20680268008141, 11.81670257138706, 12.51406531544725, 12.78349495363762, 12.96634844280359
