L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 7-s − 8-s + 9-s + 2·11-s − 12-s + 14-s + 16-s − 18-s − 4·19-s + 21-s − 2·22-s + 24-s − 27-s − 28-s − 32-s − 2·33-s + 36-s + 10·37-s + 4·38-s − 8·41-s − 42-s + 6·43-s + 2·44-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.603·11-s − 0.288·12-s + 0.267·14-s + 1/4·16-s − 0.235·18-s − 0.917·19-s + 0.218·21-s − 0.426·22-s + 0.204·24-s − 0.192·27-s − 0.188·28-s − 0.176·32-s − 0.348·33-s + 1/6·36-s + 1.64·37-s + 0.648·38-s − 1.24·41-s − 0.154·42-s + 0.914·43-s + 0.301·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 303450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 303450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8969049947\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8969049947\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 \) |
good | 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−12.80149950558890, −11.99316910469965, −11.70403535705642, −11.25524557197413, −10.96213800535855, −10.19170853397521, −9.962893286700369, −9.650863967972598, −8.925177497692728, −8.572113387044068, −8.169073332730223, −7.523848121097724, −6.970786087933627, −6.689133271005348, −6.161903308645724, −5.784168584756506, −5.132795913928721, −4.609058359398408, −3.862730015944570, −3.690885539162655, −2.685651066922622, −2.364361292036589, −1.561903968561411, −1.028537641694505, −0.3337971379470688,
0.3337971379470688, 1.028537641694505, 1.561903968561411, 2.364361292036589, 2.685651066922622, 3.690885539162655, 3.862730015944570, 4.609058359398408, 5.132795913928721, 5.784168584756506, 6.161903308645724, 6.689133271005348, 6.970786087933627, 7.523848121097724, 8.169073332730223, 8.572113387044068, 8.925177497692728, 9.650863967972598, 9.962893286700369, 10.19170853397521, 10.96213800535855, 11.25524557197413, 11.70403535705642, 11.99316910469965, 12.80149950558890