| L(s) = 1 | + 3-s + 5-s + 9-s + 4·11-s − 13-s + 15-s + 17-s − 8·19-s + 8·23-s + 25-s + 27-s − 6·29-s + 4·33-s + 6·37-s − 39-s + 6·41-s + 4·43-s + 45-s − 4·47-s − 7·49-s + 51-s + 10·53-s + 4·55-s − 8·57-s − 6·61-s − 65-s + 8·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1/3·9-s + 1.20·11-s − 0.277·13-s + 0.258·15-s + 0.242·17-s − 1.83·19-s + 1.66·23-s + 1/5·25-s + 0.192·27-s − 1.11·29-s + 0.696·33-s + 0.986·37-s − 0.160·39-s + 0.937·41-s + 0.609·43-s + 0.149·45-s − 0.583·47-s − 49-s + 0.140·51-s + 1.37·53-s + 0.539·55-s − 1.05·57-s − 0.768·61-s − 0.124·65-s + 0.963·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 212160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 212160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−13.07583325957579, −12.95387717659642, −12.47402989213534, −11.83827268544881, −11.27447210588792, −10.97015099598860, −10.38835779336822, −9.907107627064410, −9.295183525616810, −9.051339144744576, −8.719997384442320, −8.052125009395092, −7.495930018560392, −7.069979538700204, −6.450329130110867, −6.190016672368929, −5.537476169087306, −4.843812999051988, −4.388736498771148, −3.873016767578395, −3.361308826777193, −2.582140025893256, −2.267647830247826, −1.465169276745185, −1.034283171414183, 0,
1.034283171414183, 1.465169276745185, 2.267647830247826, 2.582140025893256, 3.361308826777193, 3.873016767578395, 4.388736498771148, 4.843812999051988, 5.537476169087306, 6.190016672368929, 6.450329130110867, 7.069979538700204, 7.495930018560392, 8.052125009395092, 8.719997384442320, 9.051339144744576, 9.295183525616810, 9.907107627064410, 10.38835779336822, 10.97015099598860, 11.27447210588792, 11.83827268544881, 12.47402989213534, 12.95387717659642, 13.07583325957579
