L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s − 2·7-s + 8-s + 9-s − 10-s − 4·11-s − 12-s + 4·13-s − 2·14-s + 15-s + 16-s + 18-s − 19-s − 20-s + 2·21-s − 4·22-s − 24-s + 25-s + 4·26-s − 27-s − 2·28-s − 8·29-s + 30-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 − 1.20·11-s − 0.288·12-s + 1.10·13-s − 0.534·14-s + 0.258·15-s + 1/4·16-s + 0.235·18-s − 0.229·19-s − 0.223·20-s + 0.436·21-s − 0.852·22-s − 0.204·24-s + 1/5·25-s + 0.784·26-s − 0.192·27-s − 0.377·28-s − 1.48·29-s + 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 164730 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164730 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1401842696\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1401842696\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 17 | \( 1 \) |
| 19 | \( 1 + T \) |
good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−13.15153117641715, −12.81614194994122, −12.48705949420816, −11.82248548068837, −11.31974297296890, −11.03182953194204, −10.50513030013412, −10.16122280402666, −9.417129257920018, −9.050749950322603, −8.200831738398059, −7.907977593840947, −7.364275726133856, −6.682222353436716, −6.432804800228886, −5.775543101318346, −5.366407390836467, −4.919526517929565, −4.194630101232652, −3.714165402019634, −3.240502217028737, −2.706471704462031, −1.845539291710853, −1.293361789704576, −0.09804879319088878,
0.09804879319088878, 1.293361789704576, 1.845539291710853, 2.706471704462031, 3.240502217028737, 3.714165402019634, 4.194630101232652, 4.919526517929565, 5.366407390836467, 5.775543101318346, 6.432804800228886, 6.682222353436716, 7.364275726133856, 7.907977593840947, 8.200831738398059, 9.050749950322603, 9.417129257920018, 10.16122280402666, 10.50513030013412, 11.03182953194204, 11.31974297296890, 11.82248548068837, 12.48705949420816, 12.81614194994122, 13.15153117641715