L(s) = 1 | + 2·5-s + 4·7-s − 3·9-s + 2·13-s + 17-s + 4·19-s + 4·23-s − 25-s − 6·29-s + 4·31-s + 8·35-s + 2·37-s − 6·41-s − 4·43-s − 6·45-s + 9·49-s − 6·53-s + 12·59-s + 10·61-s − 12·63-s + 4·65-s − 4·67-s − 4·71-s − 6·73-s + 12·79-s + 9·81-s + 4·83-s + ⋯ |
L(s) = 1 | + 0.894·5-s + 1.51·7-s − 9-s + 0.554·13-s + 0.242·17-s + 0.917·19-s + 0.834·23-s − 1/5·25-s − 1.11·29-s + 0.718·31-s + 1.35·35-s + 0.328·37-s − 0.937·41-s − 0.609·43-s − 0.894·45-s + 9/7·49-s − 0.824·53-s + 1.56·59-s + 1.28·61-s − 1.51·63-s + 0.496·65-s − 0.488·67-s − 0.474·71-s − 0.702·73-s + 1.35·79-s + 81-s + 0.439·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1088 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.187901169\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.187901169\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 17 | \( 1 - T \) |
good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−9.837652443494474848978581941316, −8.973843731383758445205646399185, −8.284788093614864631997648265275, −7.53071551682537672947749996043, −6.34731970371117410266133745634, −5.44649982531423407553026366598, −4.98617991507820323018626882492, −3.56448000350330022567729502783, −2.33778419746144996739254850261, −1.28551748873534205155912902202,
1.28551748873534205155912902202, 2.33778419746144996739254850261, 3.56448000350330022567729502783, 4.98617991507820323018626882492, 5.44649982531423407553026366598, 6.34731970371117410266133745634, 7.53071551682537672947749996043, 8.284788093614864631997648265275, 8.973843731383758445205646399185, 9.837652443494474848978581941316