| L(s) = 1 | + 2-s + 4-s − 5-s − 7-s + 8-s − 10-s + 13-s − 14-s + 16-s − 17-s − 7·19-s − 20-s − 3·23-s + 25-s + 26-s − 28-s + 6·29-s + 5·31-s + 32-s − 34-s + 35-s + 7·37-s − 7·38-s − 40-s + 12·41-s + 10·43-s − 3·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.377·7-s + 0.353·8-s − 0.316·10-s + 0.277·13-s − 0.267·14-s + 1/4·16-s − 0.242·17-s − 1.60·19-s − 0.223·20-s − 0.625·23-s + 1/5·25-s + 0.196·26-s − 0.188·28-s + 1.11·29-s + 0.898·31-s + 0.176·32-s − 0.171·34-s + 0.169·35-s + 1.15·37-s − 1.13·38-s − 0.158·40-s + 1.87·41-s + 1.52·43-s − 0.442·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 185130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 8 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 15 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−13.22249052166863, −12.86117115002909, −12.53726338799276, −12.12126873953060, −11.46897014351888, −11.01087168671605, −10.82748581711743, −10.04352654189002, −9.731445352701280, −9.074426042303061, −8.459298722693493, −8.093909058690139, −7.646969884496436, −6.939896441443906, −6.484491566712720, −6.056058798194925, −5.763564336357079, −4.654351628899991, −4.567948411507789, −4.048673057425773, −3.455840050000343, −2.661090805130326, −2.507159167172940, −1.579848489720805, −0.8389078048234479, 0,
0.8389078048234479, 1.579848489720805, 2.507159167172940, 2.661090805130326, 3.455840050000343, 4.048673057425773, 4.567948411507789, 4.654351628899991, 5.763564336357079, 6.056058798194925, 6.484491566712720, 6.939896441443906, 7.646969884496436, 8.093909058690139, 8.459298722693493, 9.074426042303061, 9.731445352701280, 10.04352654189002, 10.82748581711743, 11.01087168671605, 11.46897014351888, 12.12126873953060, 12.53726338799276, 12.86117115002909, 13.22249052166863
