| L(s) = 1 | − 5-s − 4·11-s − 2·13-s + 2·17-s + 4·19-s − 8·23-s + 25-s + 6·29-s − 8·31-s + 2·37-s + 2·41-s − 12·43-s + 8·47-s + 6·53-s + 4·55-s + 4·59-s − 2·61-s + 2·65-s + 12·67-s + 8·71-s + 14·73-s + 12·83-s − 2·85-s + 2·89-s − 4·95-s − 10·97-s + 101-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.20·11-s − 0.554·13-s + 0.485·17-s + 0.917·19-s − 1.66·23-s + 1/5·25-s + 1.11·29-s − 1.43·31-s + 0.328·37-s + 0.312·41-s − 1.82·43-s + 1.16·47-s + 0.824·53-s + 0.539·55-s + 0.520·59-s − 0.256·61-s + 0.248·65-s + 1.46·67-s + 0.949·71-s + 1.63·73-s + 1.31·83-s − 0.216·85-s + 0.211·89-s − 0.410·95-s − 1.01·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 141120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 141120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.348912768\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.348912768\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 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 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 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
−13.45774371366792, −12.77322282652364, −12.46166729450612, −12.01584647509879, −11.53845461612768, −11.03641854980066, −10.45312560832996, −9.982705095622132, −9.758965547150633, −9.022437669404046, −8.405535983899895, −7.942800424562725, −7.658927507536542, −7.110050251154886, −6.545155469498239, −5.851577398765235, −5.313677216305689, −5.016731488897198, −4.323678956548547, −3.584407245591358, −3.330576498701950, −2.322637713826615, −2.197808853355115, −1.074193091873716, −0.3801642269859935,
0.3801642269859935, 1.074193091873716, 2.197808853355115, 2.322637713826615, 3.330576498701950, 3.584407245591358, 4.323678956548547, 5.016731488897198, 5.313677216305689, 5.851577398765235, 6.545155469498239, 7.110050251154886, 7.658927507536542, 7.942800424562725, 8.405535983899895, 9.022437669404046, 9.758965547150633, 9.982705095622132, 10.45312560832996, 11.03641854980066, 11.53845461612768, 12.01584647509879, 12.46166729450612, 12.77322282652364, 13.45774371366792
