L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 8-s + 9-s + 4·11-s + 12-s + 2·13-s + 16-s + 3·17-s + 18-s − 19-s + 4·22-s + 2·23-s + 24-s + 2·26-s + 27-s + 2·29-s − 8·31-s + 32-s + 4·33-s + 3·34-s + 36-s + 10·37-s − 38-s + 2·39-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 + 1.20·11-s + 0.288·12-s + 0.554·13-s + 1/4·16-s + 0.727·17-s + 0.235·18-s − 0.229·19-s + 0.852·22-s + 0.417·23-s + 0.204·24-s + 0.392·26-s + 0.192·27-s + 0.371·29-s − 1.43·31-s + 0.176·32-s + 0.696·33-s + 0.514·34-s + 1/6·36-s + 1.64·37-s − 0.162·38-s + 0.320·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 139650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 139650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(8.428316541\) |
\(L(\frac12)\) |
\(\approx\) |
\(8.428316541\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 - 7 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 11 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−13.32979870961291, −12.96289482788510, −12.63201757256406, −12.06562907215758, −11.49343168946936, −11.12638850935979, −10.69531874738062, −9.982042287417313, −9.430307047684593, −9.162815748812377, −8.563688440198624, −7.893003905160505, −7.600329846510486, −6.924445957146049, −6.421643851294684, −6.027266837905050, −5.389710098232622, −4.832485033445955, −4.038343030889935, −3.866824602013003, −3.320636678621872, −2.549521342590278, −2.107095132002220, −1.213998121808412, −0.8278471509590393,
0.8278471509590393, 1.213998121808412, 2.107095132002220, 2.549521342590278, 3.320636678621872, 3.866824602013003, 4.038343030889935, 4.832485033445955, 5.389710098232622, 6.027266837905050, 6.421643851294684, 6.924445957146049, 7.600329846510486, 7.893003905160505, 8.563688440198624, 9.162815748812377, 9.430307047684593, 9.982042287417313, 10.69531874738062, 11.12638850935979, 11.49343168946936, 12.06562907215758, 12.63201757256406, 12.96289482788510, 13.32979870961291