| L(s) = 1 | − 2-s + 4-s + 2·5-s + 7-s − 8-s − 3·9-s − 2·10-s + 4·11-s − 14-s + 16-s + 5·17-s + 3·18-s + 19-s + 2·20-s − 4·22-s + 6·23-s − 25-s + 28-s − 5·29-s − 32-s − 5·34-s + 2·35-s − 3·36-s − 8·37-s − 38-s − 2·40-s − 4·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.894·5-s + 0.377·7-s − 0.353·8-s − 9-s − 0.632·10-s + 1.20·11-s − 0.267·14-s + 1/4·16-s + 1.21·17-s + 0.707·18-s + 0.229·19-s + 0.447·20-s − 0.852·22-s + 1.25·23-s − 1/5·25-s + 0.188·28-s − 0.928·29-s − 0.176·32-s − 0.857·34-s + 0.338·35-s − 1/2·36-s − 1.31·37-s − 0.162·38-s − 0.316·40-s − 0.624·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 202 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 202 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.039666595\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.039666595\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 101 | \( 1 - T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−12.14095023624894706575369889921, −11.42499939000598727705153471811, −10.37021091190968764810165266309, −9.354818220288156065213032429958, −8.708127948406865719761571922441, −7.45010683803474939843752540736, −6.24019484872868626499451155147, −5.29772327599947215266271518745, −3.27245191850330716369964188043, −1.58676544482628209078886507067,
1.58676544482628209078886507067, 3.27245191850330716369964188043, 5.29772327599947215266271518745, 6.24019484872868626499451155147, 7.45010683803474939843752540736, 8.708127948406865719761571922441, 9.354818220288156065213032429958, 10.37021091190968764810165266309, 11.42499939000598727705153471811, 12.14095023624894706575369889921
