| L(s) = 1 | + 3·3-s + 6·9-s + 3·11-s + 3·13-s + 7·17-s + 6·19-s − 6·23-s + 9·27-s + 29-s − 6·31-s + 9·33-s − 6·37-s + 9·39-s + 10·41-s + 6·43-s − 3·47-s + 21·51-s + 4·53-s + 18·57-s + 12·59-s + 6·61-s − 12·67-s − 18·69-s − 6·73-s + 15·79-s + 9·81-s + 12·83-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 2·9-s + 0.904·11-s + 0.832·13-s + 1.69·17-s + 1.37·19-s − 1.25·23-s + 1.73·27-s + 0.185·29-s − 1.07·31-s + 1.56·33-s − 0.986·37-s + 1.44·39-s + 1.56·41-s + 0.914·43-s − 0.437·47-s + 2.94·51-s + 0.549·53-s + 2.38·57-s + 1.56·59-s + 0.768·61-s − 1.46·67-s − 2.16·69-s − 0.702·73-s + 1.68·79-s + 81-s + 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.723998568\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.723998568\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| good | 3 | \( 1 - p T + p T^{2} \) | 1.3.ad |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 - 3 T + p T^{2} \) | 1.13.ad |
| 17 | \( 1 - 7 T + p T^{2} \) | 1.17.ah |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - T + p T^{2} \) | 1.29.ab |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 15 T + p T^{2} \) | 1.79.ap |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 + 9 T + p T^{2} \) | 1.97.j |
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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
−14.11568583775199, −13.74719567668665, −13.16251029901669, −12.64313865762458, −11.97283816243650, −11.80976078346285, −10.89403262866980, −10.36493986059835, −9.677494564112486, −9.554351104220157, −8.904717315658676, −8.529282482157835, −7.812886047890655, −7.621297822089183, −7.074910604786903, −6.320738885959874, −5.681571313865891, −5.185762763765272, −4.099254000010801, −3.835439306250707, −3.400903290097018, −2.806369558947502, −2.058065596902063, −1.396594784101003, −0.8925914548537625,
0.8925914548537625, 1.396594784101003, 2.058065596902063, 2.806369558947502, 3.400903290097018, 3.835439306250707, 4.099254000010801, 5.185762763765272, 5.681571313865891, 6.320738885959874, 7.074910604786903, 7.621297822089183, 7.812886047890655, 8.529282482157835, 8.904717315658676, 9.554351104220157, 9.677494564112486, 10.36493986059835, 10.89403262866980, 11.80976078346285, 11.97283816243650, 12.64313865762458, 13.16251029901669, 13.74719567668665, 14.11568583775199
