| L(s) = 1 | + 3-s − 7-s + 9-s + 4·11-s + 2·13-s − 2·17-s − 4·19-s − 21-s − 8·23-s + 27-s − 2·29-s + 4·33-s − 6·37-s + 2·39-s − 6·41-s − 4·43-s + 49-s − 2·51-s + 10·53-s − 4·57-s − 12·59-s + 14·61-s − 63-s − 12·67-s − 8·69-s + 8·71-s − 10·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.377·7-s + 1/3·9-s + 1.20·11-s + 0.554·13-s − 0.485·17-s − 0.917·19-s − 0.218·21-s − 1.66·23-s + 0.192·27-s − 0.371·29-s + 0.696·33-s − 0.986·37-s + 0.320·39-s − 0.937·41-s − 0.609·43-s + 1/7·49-s − 0.280·51-s + 1.37·53-s − 0.529·57-s − 1.56·59-s + 1.79·61-s − 0.125·63-s − 1.46·67-s − 0.963·69-s + 0.949·71-s − 1.17·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 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 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−7.43111101448414744645694371828, −6.68401139093363229684792093128, −6.26943055353698692847569199410, −5.45652873078353554377952793071, −4.29143786981852147115199734824, −3.94840504042136705145616607456, −3.18406525330673463656651412069, −2.12597879324065521917847731537, −1.47070780323706851028293863910, 0,
1.47070780323706851028293863910, 2.12597879324065521917847731537, 3.18406525330673463656651412069, 3.94840504042136705145616607456, 4.29143786981852147115199734824, 5.45652873078353554377952793071, 6.26943055353698692847569199410, 6.68401139093363229684792093128, 7.43111101448414744645694371828
