L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 3·7-s − 8-s + 9-s + 10-s + 11-s − 12-s − 2·13-s + 3·14-s + 15-s + 16-s + 17-s − 18-s − 4·19-s − 20-s + 3·21-s − 22-s + 9·23-s + 24-s − 4·25-s + 2·26-s − 27-s − 3·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 1.13·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.301·11-s − 0.288·12-s − 0.554·13-s + 0.801·14-s + 0.258·15-s + 1/4·16-s + 0.242·17-s − 0.235·18-s − 0.917·19-s − 0.223·20-s + 0.654·21-s − 0.213·22-s + 1.87·23-s + 0.204·24-s − 4/5·25-s + 0.392·26-s − 0.192·27-s − 0.566·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 786 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 786 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6243670998\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6243670998\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 131 | \( 1 - T \) |
good | 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 5 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−10.24735782349002009516889838972, −9.509137063633831977341435711423, −8.720642353869452303722670476388, −7.65209544587413833911721371705, −6.79486751320802487244077035076, −6.21648019265830441227921123018, −4.97177722210454931101260498044, −3.75214105042936225291187186706, −2.57028375210575225885074827625, −0.72438559363197131164781371675,
0.72438559363197131164781371675, 2.57028375210575225885074827625, 3.75214105042936225291187186706, 4.97177722210454931101260498044, 6.21648019265830441227921123018, 6.79486751320802487244077035076, 7.65209544587413833911721371705, 8.720642353869452303722670476388, 9.509137063633831977341435711423, 10.24735782349002009516889838972