| L(s) = 1 | + 2·2-s + 4·4-s − 5·5-s + 14·7-s + 8·8-s − 10·10-s − 3·11-s + 47·13-s + 28·14-s + 16·16-s + 39·17-s + 32·19-s − 20·20-s − 6·22-s + 99·23-s + 25·25-s + 94·26-s + 56·28-s − 51·29-s + 83·31-s + 32·32-s + 78·34-s − 70·35-s + 314·37-s + 64·38-s − 40·40-s + 108·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.755·7-s + 0.353·8-s − 0.316·10-s − 0.0822·11-s + 1.00·13-s + 0.534·14-s + 1/4·16-s + 0.556·17-s + 0.386·19-s − 0.223·20-s − 0.0581·22-s + 0.897·23-s + 1/5·25-s + 0.709·26-s + 0.377·28-s − 0.326·29-s + 0.480·31-s + 0.176·32-s + 0.393·34-s − 0.338·35-s + 1.39·37-s + 0.273·38-s − 0.158·40-s + 0.411·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.014685092\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.014685092\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + p T \) |
| good | 7 | \( 1 - 2 p T + p^{3} T^{2} \) |
| 11 | \( 1 + 3 T + p^{3} T^{2} \) |
| 13 | \( 1 - 47 T + p^{3} T^{2} \) |
| 17 | \( 1 - 39 T + p^{3} T^{2} \) |
| 19 | \( 1 - 32 T + p^{3} T^{2} \) |
| 23 | \( 1 - 99 T + p^{3} T^{2} \) |
| 29 | \( 1 + 51 T + p^{3} T^{2} \) |
| 31 | \( 1 - 83 T + p^{3} T^{2} \) |
| 37 | \( 1 - 314 T + p^{3} T^{2} \) |
| 41 | \( 1 - 108 T + p^{3} T^{2} \) |
| 43 | \( 1 - 299 T + p^{3} T^{2} \) |
| 47 | \( 1 + 531 T + p^{3} T^{2} \) |
| 53 | \( 1 + 564 T + p^{3} T^{2} \) |
| 59 | \( 1 + 12 T + p^{3} T^{2} \) |
| 61 | \( 1 - 230 T + p^{3} T^{2} \) |
| 67 | \( 1 + 4 p T + p^{3} T^{2} \) |
| 71 | \( 1 + 120 T + p^{3} T^{2} \) |
| 73 | \( 1 - 1106 T + p^{3} T^{2} \) |
| 79 | \( 1 + 739 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1086 T + p^{3} T^{2} \) |
| 89 | \( 1 - 120 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1642 T + p^{3} T^{2} \) |
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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
−11.34106330990542563348986185600, −11.03681396601531768652904083476, −9.641447202588911584911439410560, −8.346704299076133895999256290881, −7.58292040328467036865502877057, −6.35553036251532870381636664131, −5.23895334946413519346716508743, −4.20292216471389383059227878191, −3.00896933479992684974418391341, −1.27242072369560111817595953244,
1.27242072369560111817595953244, 3.00896933479992684974418391341, 4.20292216471389383059227878191, 5.23895334946413519346716508743, 6.35553036251532870381636664131, 7.58292040328467036865502877057, 8.346704299076133895999256290881, 9.641447202588911584911439410560, 11.03681396601531768652904083476, 11.34106330990542563348986185600
