L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 2·7-s − 8-s + 9-s − 10-s − 12-s + 2·14-s − 15-s + 16-s − 17-s − 18-s − 4·19-s + 20-s + 2·21-s + 4·23-s + 24-s + 25-s − 27-s − 2·28-s + 2·29-s + 30-s − 32-s + 34-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.755·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.288·12-s + 0.534·14-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 + 0.436·21-s + 0.834·23-s + 0.204·24-s + 1/5·25-s − 0.192·27-s − 0.377·28-s + 0.371·29-s + 0.182·30-s − 0.176·32-s + 0.171·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 86190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 86190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.073961882\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.073961882\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 \) |
| 17 | \( 1 + T \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 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
−13.86941534409713, −13.24140294036984, −12.92094162071520, −12.27547365137673, −12.11055227631500, −11.09145304863419, −10.96657838696849, −10.48535180080175, −9.868156123670858, −9.470695607113220, −8.953986201156097, −8.564349424805281, −7.739230716054679, −7.316686853491154, −6.658917152097452, −6.344661294133432, −5.799459704352846, −5.293541179109108, −4.435848280728861, −4.047922323077936, −3.057696183163841, −2.633659917981445, −1.902890533553462, −1.101511876631257, −0.4342870498408680,
0.4342870498408680, 1.101511876631257, 1.902890533553462, 2.633659917981445, 3.057696183163841, 4.047922323077936, 4.435848280728861, 5.293541179109108, 5.799459704352846, 6.344661294133432, 6.658917152097452, 7.316686853491154, 7.739230716054679, 8.564349424805281, 8.953986201156097, 9.470695607113220, 9.868156123670858, 10.48535180080175, 10.96657838696849, 11.09145304863419, 12.11055227631500, 12.27547365137673, 12.92094162071520, 13.24140294036984, 13.86941534409713