L(s) = 1 | + 3-s − 5-s − 2·9-s − 11-s − 5·13-s − 15-s + 17-s − 6·19-s − 4·23-s + 25-s − 5·27-s + 3·29-s + 2·31-s − 33-s + 8·37-s − 5·39-s − 10·41-s − 2·43-s + 2·45-s − 7·47-s + 51-s − 2·53-s + 55-s − 6·57-s + 14·59-s − 8·61-s + 5·65-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s − 2/3·9-s − 0.301·11-s − 1.38·13-s − 0.258·15-s + 0.242·17-s − 1.37·19-s − 0.834·23-s + 1/5·25-s − 0.962·27-s + 0.557·29-s + 0.359·31-s − 0.174·33-s + 1.31·37-s − 0.800·39-s − 1.56·41-s − 0.304·43-s + 0.298·45-s − 1.02·47-s + 0.140·51-s − 0.274·53-s + 0.134·55-s − 0.794·57-s + 1.82·59-s − 1.02·61-s + 0.620·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 + 3 T + p T^{2} \) |
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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
−9.656499675068027474245097738297, −8.477607699734063623594644331341, −8.153826744203389896658995586752, −7.18283360326914276923908828855, −6.21140679999298824502923361722, −5.10138965377572553769121334401, −4.18418646376772260781956869323, −3.02440876476817737528832980171, −2.16393351624843136438832605769, 0,
2.16393351624843136438832605769, 3.02440876476817737528832980171, 4.18418646376772260781956869323, 5.10138965377572553769121334401, 6.21140679999298824502923361722, 7.18283360326914276923908828855, 8.153826744203389896658995586752, 8.477607699734063623594644331341, 9.656499675068027474245097738297