| L(s) = 1 | + 3-s − 5-s − 7-s + 9-s + 3·11-s + 13-s − 15-s + 3·17-s + 2·19-s − 21-s + 3·23-s + 25-s + 27-s + 6·29-s + 2·31-s + 3·33-s + 35-s − 7·37-s + 39-s + 9·41-s + 8·43-s − 45-s − 6·47-s − 6·49-s + 3·51-s + 3·53-s − 3·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s + 0.904·11-s + 0.277·13-s − 0.258·15-s + 0.727·17-s + 0.458·19-s − 0.218·21-s + 0.625·23-s + 1/5·25-s + 0.192·27-s + 1.11·29-s + 0.359·31-s + 0.522·33-s + 0.169·35-s − 1.15·37-s + 0.160·39-s + 1.40·41-s + 1.21·43-s − 0.149·45-s − 0.875·47-s − 6/7·49-s + 0.420·51-s + 0.412·53-s − 0.404·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 780 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 780 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.839012564\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.839012564\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 5 | \( 1 + T \) | |
| 13 | \( 1 - T \) | |
| good | 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 - 9 T + p T^{2} \) | 1.41.aj |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 7 T + p T^{2} \) | 1.61.h |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 3 T + p T^{2} \) | 1.71.ad |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + T + p T^{2} \) | 1.79.b |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 3 T + p T^{2} \) | 1.89.ad |
| 97 | \( 1 + T + p T^{2} \) | 1.97.b |
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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
−10.18606145016583397681105528363, −9.353300675060649267457972418752, −8.667799177912496776637477531744, −7.75232112169166988797835867308, −6.93472752226523151674762451721, −6.01294954455979692960592001952, −4.71962747561514045447774248368, −3.70935237425456034212696337553, −2.87349232428383310425217161031, −1.20374375117958900178374083459,
1.20374375117958900178374083459, 2.87349232428383310425217161031, 3.70935237425456034212696337553, 4.71962747561514045447774248368, 6.01294954455979692960592001952, 6.93472752226523151674762451721, 7.75232112169166988797835867308, 8.667799177912496776637477531744, 9.353300675060649267457972418752, 10.18606145016583397681105528363
