| L(s) = 1 | + 2-s + 4-s − 5-s − 2·7-s + 8-s − 10-s − 3·11-s + 13-s − 2·14-s + 16-s − 19-s − 20-s − 3·22-s − 3·23-s + 25-s + 26-s − 2·28-s − 3·29-s − 5·31-s + 32-s + 2·35-s + 7·37-s − 38-s − 40-s − 43-s − 3·44-s − 3·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.755·7-s + 0.353·8-s − 0.316·10-s − 0.904·11-s + 0.277·13-s − 0.534·14-s + 1/4·16-s − 0.229·19-s − 0.223·20-s − 0.639·22-s − 0.625·23-s + 1/5·25-s + 0.196·26-s − 0.377·28-s − 0.557·29-s − 0.898·31-s + 0.176·32-s + 0.338·35-s + 1.15·37-s − 0.162·38-s − 0.158·40-s − 0.152·43-s − 0.452·44-s − 0.442·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 338130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 17 T + p T^{2} \) |
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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
−13.02837005929892, −12.62823773905333, −12.30252263821298, −11.69150664367182, −11.24650033630284, −10.84813612879652, −10.46286573959570, −9.835625739524892, −9.513340900625356, −8.931281411016288, −8.324830763296139, −7.871094430914248, −7.501535367626855, −6.982867972847626, −6.443935437435523, −5.953693186030725, −5.621982321342194, −5.012151475791333, −4.427317924625365, −4.042140173943166, −3.469971372612670, −2.936288682705959, −2.602371478148326, −1.794108251025410, −1.238702779092883, 0, 0,
1.238702779092883, 1.794108251025410, 2.602371478148326, 2.936288682705959, 3.469971372612670, 4.042140173943166, 4.427317924625365, 5.012151475791333, 5.621982321342194, 5.953693186030725, 6.443935437435523, 6.982867972847626, 7.501535367626855, 7.871094430914248, 8.324830763296139, 8.931281411016288, 9.513340900625356, 9.835625739524892, 10.46286573959570, 10.84813612879652, 11.24650033630284, 11.69150664367182, 12.30252263821298, 12.62823773905333, 13.02837005929892
