| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 8-s + 9-s − 12-s + 16-s + 6·17-s + 18-s + 6·19-s + 6·23-s − 24-s − 5·25-s − 27-s − 6·29-s − 4·31-s + 32-s + 6·34-s + 36-s + 2·37-s + 6·38-s + 6·41-s − 6·43-s + 6·46-s − 6·47-s − 48-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.288·12-s + 1/4·16-s + 1.45·17-s + 0.235·18-s + 1.37·19-s + 1.25·23-s − 0.204·24-s − 25-s − 0.192·27-s − 1.11·29-s − 0.718·31-s + 0.176·32-s + 1.02·34-s + 1/6·36-s + 0.328·37-s + 0.973·38-s + 0.937·41-s − 0.914·43-s + 0.884·46-s − 0.875·47-s − 0.144·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 122694 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122694 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.402779080\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.402779080\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 3 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 + 12 T + p T^{2} \) | 1.73.m |
| 79 | \( 1 + 12 T + p T^{2} \) | 1.79.m |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−13.33026641283206, −13.11388995283929, −12.62829030661478, −12.08826883112087, −11.59406983593157, −11.27179488422141, −10.90955616980067, −9.994119004242226, −9.865383675719430, −9.324590812588495, −8.626861936939308, −7.828886466037602, −7.491736089135010, −7.180285063150491, −6.367991569549732, −5.901728345689847, −5.467785946298453, −5.008662533576400, −4.537141154336136, −3.630580643996264, −3.379010689151452, −2.797904630409000, −1.773983156084523, −1.371952686602016, −0.5256889762781310,
0.5256889762781310, 1.371952686602016, 1.773983156084523, 2.797904630409000, 3.379010689151452, 3.630580643996264, 4.537141154336136, 5.008662533576400, 5.467785946298453, 5.901728345689847, 6.367991569549732, 7.180285063150491, 7.491736089135010, 7.828886466037602, 8.626861936939308, 9.324590812588495, 9.865383675719430, 9.994119004242226, 10.90955616980067, 11.27179488422141, 11.59406983593157, 12.08826883112087, 12.62829030661478, 13.11388995283929, 13.33026641283206
