| L(s) = 1 | + 2-s + 4-s − 5-s − 4.89·7-s + 8-s − 10-s + 6.89·13-s − 4.89·14-s + 16-s + 17-s + 4·19-s − 20-s + 4·23-s + 25-s + 6.89·26-s − 4.89·28-s − 6·29-s + 4·31-s + 32-s + 34-s + 4.89·35-s + 6·37-s + 4·38-s − 40-s + 2.89·41-s + 8.89·43-s + 4·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.447·5-s − 1.85·7-s + 0.353·8-s − 0.316·10-s + 1.91·13-s − 1.30·14-s + 0.250·16-s + 0.242·17-s + 0.917·19-s − 0.223·20-s + 0.834·23-s + 0.200·25-s + 1.35·26-s − 0.925·28-s − 1.11·29-s + 0.718·31-s + 0.176·32-s + 0.171·34-s + 0.828·35-s + 0.986·37-s + 0.648·38-s − 0.158·40-s + 0.452·41-s + 1.35·43-s + 0.589·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1530 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1530 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.193313429\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.193313429\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 + 4.89T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 6.89T + 13T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 - 2.89T + 41T^{2} \) |
| 43 | \( 1 - 8.89T + 43T^{2} \) |
| 47 | \( 1 + 9.79T + 47T^{2} \) |
| 53 | \( 1 - 7.79T + 53T^{2} \) |
| 59 | \( 1 + 4.89T + 59T^{2} \) |
| 61 | \( 1 - 11.7T + 61T^{2} \) |
| 67 | \( 1 - 0.898T + 67T^{2} \) |
| 71 | \( 1 - 8.89T + 71T^{2} \) |
| 73 | \( 1 + 10.8T + 73T^{2} \) |
| 79 | \( 1 + 5.79T + 79T^{2} \) |
| 83 | \( 1 + 13.7T + 83T^{2} \) |
| 89 | \( 1 - 7.79T + 89T^{2} \) |
| 97 | \( 1 + 12.6T + 97T^{2} \) |
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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
−9.471003516887118096789787385385, −8.748758402490878670715173846308, −7.67169597024086905553929154563, −6.83613441318263370147711341573, −6.14855229390493319182489238105, −5.52453065089165432788118151467, −4.11504912211704662616975498887, −3.49693096062779341182140934798, −2.81352376324471588332664228962, −0.972619348093426944302437554679,
0.972619348093426944302437554679, 2.81352376324471588332664228962, 3.49693096062779341182140934798, 4.11504912211704662616975498887, 5.52453065089165432788118151467, 6.14855229390493319182489238105, 6.83613441318263370147711341573, 7.67169597024086905553929154563, 8.748758402490878670715173846308, 9.471003516887118096789787385385
