| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 8-s + 9-s − 3·11-s + 12-s + 5·13-s + 16-s − 17-s + 18-s − 2·19-s − 3·22-s + 24-s + 5·26-s + 27-s + 6·29-s − 5·31-s + 32-s − 3·33-s − 34-s + 36-s − 2·37-s − 2·38-s + 5·39-s + 4·43-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 − 0.904·11-s + 0.288·12-s + 1.38·13-s + 1/4·16-s − 0.242·17-s + 0.235·18-s − 0.458·19-s − 0.639·22-s + 0.204·24-s + 0.980·26-s + 0.192·27-s + 1.11·29-s − 0.898·31-s + 0.176·32-s − 0.522·33-s − 0.171·34-s + 1/6·36-s − 0.328·37-s − 0.324·38-s + 0.800·39-s + 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 124950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.778259931\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.778259931\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| 17 | \( 1 + T \) | |
| good | 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 - 5 T + p T^{2} \) | 1.13.af |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 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.c |
| 71 | \( 1 - 3 T + p T^{2} \) | 1.71.ad |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 15 T + p T^{2} \) | 1.89.p |
| 97 | \( 1 + 16 T + p T^{2} \) | 1.97.q |
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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
−13.67461578169808, −13.06959805990695, −12.69386541428301, −12.28730478941617, −11.62196853361625, −11.01190048121942, −10.75115835676779, −10.26764137235334, −9.675406020617034, −9.020488393067574, −8.451865310459928, −8.278056625898777, −7.543519927660369, −7.041463952733278, −6.537025951843997, −5.948512760586530, −5.452449431739726, −4.930516957112261, −4.186117391320948, −3.833087101947074, −3.272053233454606, −2.551900537448093, −2.197985029909881, −1.364375106567262, −0.6246174718203466,
0.6246174718203466, 1.364375106567262, 2.197985029909881, 2.551900537448093, 3.272053233454606, 3.833087101947074, 4.186117391320948, 4.930516957112261, 5.452449431739726, 5.948512760586530, 6.537025951843997, 7.041463952733278, 7.543519927660369, 8.278056625898777, 8.451865310459928, 9.020488393067574, 9.675406020617034, 10.26764137235334, 10.75115835676779, 11.01190048121942, 11.62196853361625, 12.28730478941617, 12.69386541428301, 13.06959805990695, 13.67461578169808
