L(s) = 1 | − 2-s + 4-s − 7-s − 8-s + 6·13-s + 14-s + 16-s + 6·19-s − 23-s − 6·26-s − 28-s − 4·31-s − 32-s + 10·37-s − 6·38-s + 6·41-s − 4·43-s + 46-s + 49-s + 6·52-s − 10·53-s + 56-s + 4·59-s + 4·62-s + 64-s − 2·71-s − 2·73-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s + 1.66·13-s + 0.267·14-s + 1/4·16-s + 1.37·19-s − 0.208·23-s − 1.17·26-s − 0.188·28-s − 0.718·31-s − 0.176·32-s + 1.64·37-s − 0.973·38-s + 0.937·41-s − 0.609·43-s + 0.147·46-s + 1/7·49-s + 0.832·52-s − 1.37·53-s + 0.133·56-s + 0.520·59-s + 0.508·62-s + 1/8·64-s − 0.237·71-s − 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.942036926\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.942036926\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 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
−14.15615545641194, −13.58921547539000, −12.96735914522285, −12.77938164582695, −11.89960413107313, −11.48723215515410, −11.07904841758100, −10.63189272351505, −9.914332452746661, −9.548447061164874, −9.085311967250384, −8.509571323650276, −8.013567949698381, −7.493157162696387, −6.975621349068312, −6.223166721974591, −5.968253469211772, −5.364125599973676, −4.521663632076665, −3.836112593895015, −3.279907873318964, −2.768223650627030, −1.839701207923373, −1.201561904254499, −0.5764523486128935,
0.5764523486128935, 1.201561904254499, 1.839701207923373, 2.768223650627030, 3.279907873318964, 3.836112593895015, 4.521663632076665, 5.364125599973676, 5.968253469211772, 6.223166721974591, 6.975621349068312, 7.493157162696387, 8.013567949698381, 8.509571323650276, 9.085311967250384, 9.548447061164874, 9.914332452746661, 10.63189272351505, 11.07904841758100, 11.48723215515410, 11.89960413107313, 12.77938164582695, 12.96735914522285, 13.58921547539000, 14.15615545641194