| L(s) = 1 | + 5-s + 11-s − 13-s + 6·17-s − 4·19-s + 8·23-s + 25-s − 6·29-s − 4·31-s − 2·37-s + 6·41-s − 4·43-s − 7·49-s − 10·53-s + 55-s + 12·59-s − 6·61-s − 65-s − 4·67-s − 6·73-s − 8·79-s + 12·83-s + 6·85-s − 6·89-s − 4·95-s + 18·97-s + 101-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.301·11-s − 0.277·13-s + 1.45·17-s − 0.917·19-s + 1.66·23-s + 1/5·25-s − 1.11·29-s − 0.718·31-s − 0.328·37-s + 0.937·41-s − 0.609·43-s − 49-s − 1.37·53-s + 0.134·55-s + 1.56·59-s − 0.768·61-s − 0.124·65-s − 0.488·67-s − 0.702·73-s − 0.900·79-s + 1.31·83-s + 0.650·85-s − 0.635·89-s − 0.410·95-s + 1.82·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 102960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 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 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−14.15365537656579, −13.24821498192193, −13.02500826727669, −12.66736598310547, −12.02271085193924, −11.52261611422555, −10.98067352397383, −10.57767325313990, −10.00607404963175, −9.484220668370986, −9.114793024401444, −8.590322827835509, −7.947246998083922, −7.388387612334396, −7.031927169119797, −6.280861021478486, −5.917023794565136, −5.208604597477152, −4.898034494449134, −4.134311448310157, −3.418282344376208, −3.065799409071102, −2.221878170969206, −1.607168082312414, −0.9795895853319597, 0,
0.9795895853319597, 1.607168082312414, 2.221878170969206, 3.065799409071102, 3.418282344376208, 4.134311448310157, 4.898034494449134, 5.208604597477152, 5.917023794565136, 6.280861021478486, 7.031927169119797, 7.388387612334396, 7.947246998083922, 8.590322827835509, 9.114793024401444, 9.484220668370986, 10.00607404963175, 10.57767325313990, 10.98067352397383, 11.52261611422555, 12.02271085193924, 12.66736598310547, 13.02500826727669, 13.24821498192193, 14.15365537656579
