L(s) = 1 | − 2-s − 3-s + 4-s + 2·5-s + 6-s + 4·7-s − 8-s − 2·9-s − 2·10-s − 2·11-s − 12-s − 7·13-s − 4·14-s − 2·15-s + 16-s + 17-s + 2·18-s + 8·19-s + 2·20-s − 4·21-s + 2·22-s − 5·23-s + 24-s − 25-s + 7·26-s + 5·27-s + 4·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.894·5-s + 0.408·6-s + 1.51·7-s − 0.353·8-s − 2/3·9-s − 0.632·10-s − 0.603·11-s − 0.288·12-s − 1.94·13-s − 1.06·14-s − 0.516·15-s + 1/4·16-s + 0.242·17-s + 0.471·18-s + 1.83·19-s + 0.447·20-s − 0.872·21-s + 0.426·22-s − 1.04·23-s + 0.204·24-s − 1/5·25-s + 1.37·26-s + 0.962·27-s + 0.755·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.229353415\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.229353415\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 + T \) |
good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 - 5 T + p T^{2} \) |
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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
−9.273561321596953755350316980190, −8.214102783432576300212946049496, −7.78035097360251725695664749751, −6.95226757091664512009378526795, −5.79219994929784060993031593515, −5.29522997287902234900825748237, −4.65570254380202185042458656598, −2.79476325351105333240332740001, −2.13004012112302686616500178251, −0.848974027129899434187830120264,
0.848974027129899434187830120264, 2.13004012112302686616500178251, 2.79476325351105333240332740001, 4.65570254380202185042458656598, 5.29522997287902234900825748237, 5.79219994929784060993031593515, 6.95226757091664512009378526795, 7.78035097360251725695664749751, 8.214102783432576300212946049496, 9.273561321596953755350316980190