| L(s) = 1 | − 7-s − 4·11-s − 7·13-s − 17-s + 6·19-s − 8·23-s + 7·31-s + 8·37-s + 6·41-s − 4·43-s − 6·47-s − 6·49-s + 3·53-s − 12·59-s + 14·61-s − 8·67-s − 9·71-s + 4·73-s + 4·77-s − 5·79-s + 4·83-s + 6·89-s + 7·91-s − 8·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 0.377·7-s − 1.20·11-s − 1.94·13-s − 0.242·17-s + 1.37·19-s − 1.66·23-s + 1.25·31-s + 1.31·37-s + 0.937·41-s − 0.609·43-s − 0.875·47-s − 6/7·49-s + 0.412·53-s − 1.56·59-s + 1.79·61-s − 0.977·67-s − 1.06·71-s + 0.468·73-s + 0.455·77-s − 0.562·79-s + 0.439·83-s + 0.635·89-s + 0.733·91-s − 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 244800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 244800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5731253490\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5731253490\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 17 | \( 1 + T \) | |
| good | 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 + 7 T + p T^{2} \) | 1.13.h |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 7 T + p T^{2} \) | 1.31.ah |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + 9 T + p T^{2} \) | 1.71.j |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 + 5 T + p T^{2} \) | 1.79.f |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−12.92005428807238, −12.21665019446422, −12.15035466444427, −11.51619981690202, −11.12471234023223, −10.33612734127768, −9.980038742519455, −9.763475795444336, −9.390242843661040, −8.568415798818659, −8.020404846410324, −7.697226055306728, −7.348290912104730, −6.762331218048561, −6.073961537277522, −5.764272150199291, −5.056492254562635, −4.713603576505090, −4.281017755065194, −3.394938339776415, −2.916954772140041, −2.455623951198934, −1.994943820870303, −1.039557933051385, −0.2218918020811365,
0.2218918020811365, 1.039557933051385, 1.994943820870303, 2.455623951198934, 2.916954772140041, 3.394938339776415, 4.281017755065194, 4.713603576505090, 5.056492254562635, 5.764272150199291, 6.073961537277522, 6.762331218048561, 7.348290912104730, 7.697226055306728, 8.020404846410324, 8.568415798818659, 9.390242843661040, 9.763475795444336, 9.980038742519455, 10.33612734127768, 11.12471234023223, 11.51619981690202, 12.15035466444427, 12.21665019446422, 12.92005428807238
