| L(s) = 1 | + 2-s − 2·3-s + 4-s − 5-s − 2·6-s + 8-s + 9-s − 10-s − 4·11-s − 2·12-s − 2·13-s + 2·15-s + 16-s − 8·17-s + 18-s + 6·19-s − 20-s − 4·22-s − 4·23-s − 2·24-s + 25-s − 2·26-s + 4·27-s − 6·29-s + 2·30-s − 4·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.447·5-s − 0.816·6-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 1.20·11-s − 0.577·12-s − 0.554·13-s + 0.516·15-s + 1/4·16-s − 1.94·17-s + 0.235·18-s + 1.37·19-s − 0.223·20-s − 0.852·22-s − 0.834·23-s − 0.408·24-s + 1/5·25-s − 0.392·26-s + 0.769·27-s − 1.11·29-s + 0.365·30-s − 0.718·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−10.86614564966969779050895065772, −9.987626462691323536290955576580, −8.621807622472038281841661682648, −7.43586792817004186337731069491, −6.72584879403071822637349996832, −5.47531989167733320735048693294, −5.07475421442181782846073138217, −3.82742742653431763007701470093, −2.35089027266828839682953321286, 0,
2.35089027266828839682953321286, 3.82742742653431763007701470093, 5.07475421442181782846073138217, 5.47531989167733320735048693294, 6.72584879403071822637349996832, 7.43586792817004186337731069491, 8.621807622472038281841661682648, 9.987626462691323536290955576580, 10.86614564966969779050895065772
