| L(s) = 1 | − 3-s − 2·9-s − 3·11-s + 2·13-s − 17-s + 4·19-s + 5·27-s − 5·31-s + 3·33-s + 2·37-s − 2·39-s − 8·43-s + 6·47-s + 51-s + 6·53-s − 4·57-s + 8·61-s − 8·67-s − 4·73-s + 79-s + 81-s − 6·83-s − 3·89-s + 5·93-s − 10·97-s + 6·99-s + 101-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 2/3·9-s − 0.904·11-s + 0.554·13-s − 0.242·17-s + 0.917·19-s + 0.962·27-s − 0.898·31-s + 0.522·33-s + 0.328·37-s − 0.320·39-s − 1.21·43-s + 0.875·47-s + 0.140·51-s + 0.824·53-s − 0.529·57-s + 1.02·61-s − 0.977·67-s − 0.468·73-s + 0.112·79-s + 1/9·81-s − 0.658·83-s − 0.317·89-s + 0.518·93-s − 1.01·97-s + 0.603·99-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 333200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 333200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−12.91021813932379, −12.25127888615434, −11.83505166853171, −11.49462347576056, −10.92149755671544, −10.76922767881360, −10.07896056512642, −9.786029219274647, −9.111452325474105, −8.558465263052090, −8.420443146631237, −7.647508781492891, −7.286514079889073, −6.811165334308488, −6.101458554163674, −5.810812456750746, −5.330341110487291, −4.955029078076246, −4.347175051329198, −3.665053701759407, −3.194834528762581, −2.652043481594961, −2.083836187034256, −1.321584062495509, −0.6538972271463817, 0,
0.6538972271463817, 1.321584062495509, 2.083836187034256, 2.652043481594961, 3.194834528762581, 3.665053701759407, 4.347175051329198, 4.955029078076246, 5.330341110487291, 5.810812456750746, 6.101458554163674, 6.811165334308488, 7.286514079889073, 7.647508781492891, 8.420443146631237, 8.558465263052090, 9.111452325474105, 9.786029219274647, 10.07896056512642, 10.76922767881360, 10.92149755671544, 11.49462347576056, 11.83505166853171, 12.25127888615434, 12.91021813932379
