| L(s) = 1 | + 3-s + 9-s + 11-s − 2·13-s + 4·17-s + 2·19-s + 27-s − 4·29-s − 2·31-s + 33-s + 37-s − 2·39-s + 10·41-s + 2·43-s + 8·47-s − 7·49-s + 4·51-s − 10·53-s + 2·57-s − 10·61-s + 8·67-s + 16·71-s − 10·73-s − 14·79-s + 81-s − 4·83-s − 4·87-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s + 0.301·11-s − 0.554·13-s + 0.970·17-s + 0.458·19-s + 0.192·27-s − 0.742·29-s − 0.359·31-s + 0.174·33-s + 0.164·37-s − 0.320·39-s + 1.56·41-s + 0.304·43-s + 1.16·47-s − 49-s + 0.560·51-s − 1.37·53-s + 0.264·57-s − 1.28·61-s + 0.977·67-s + 1.89·71-s − 1.17·73-s − 1.57·79-s + 1/9·81-s − 0.439·83-s − 0.428·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 122100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| 37 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 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
−14.01121075136728, −13.30455286553817, −12.73858646228861, −12.45383299328760, −11.98965888361136, −11.25175034750431, −11.00205347600156, −10.30034265938365, −9.773464773283802, −9.317595996969463, −9.147364483015754, −8.219513141070941, −7.930920037850467, −7.405052066981211, −6.995485011378271, −6.310320801244033, −5.654588552367880, −5.333103193032391, −4.484225621163417, −4.116508704078703, −3.398389893222662, −2.933798098494437, −2.327994647336844, −1.571863712139596, −1.004074895433359, 0,
1.004074895433359, 1.571863712139596, 2.327994647336844, 2.933798098494437, 3.398389893222662, 4.116508704078703, 4.484225621163417, 5.333103193032391, 5.654588552367880, 6.310320801244033, 6.995485011378271, 7.405052066981211, 7.930920037850467, 8.219513141070941, 9.147364483015754, 9.317595996969463, 9.773464773283802, 10.30034265938365, 11.00205347600156, 11.25175034750431, 11.98965888361136, 12.45383299328760, 12.73858646228861, 13.30455286553817, 14.01121075136728
