| L(s) = 1 | + 2·2-s + 3.01·3-s + 4·4-s + 6.03·6-s + 4.64·7-s + 8·8-s + 0.103·9-s + 12.0·12-s + 9.29·14-s + 16·16-s + 0.206·18-s + 14.0·21-s + 27.7·23-s + 24.1·24-s − 26.8·27-s + 18.5·28-s − 57.8·29-s + 32·32-s + 0.413·36-s + 70.1·41-s + 28.0·42-s − 14.7·43-s + 55.4·46-s − 88.0·47-s + 48.2·48-s − 27.3·49-s − 53.6·54-s + ⋯ |
| L(s) = 1 | + 2-s + 1.00·3-s + 4-s + 1.00·6-s + 0.664·7-s + 8-s + 0.0114·9-s + 1.00·12-s + 0.664·14-s + 16-s + 0.0114·18-s + 0.668·21-s + 1.20·23-s + 1.00·24-s − 0.994·27-s + 0.664·28-s − 1.99·29-s + 32-s + 0.0114·36-s + 1.71·41-s + 0.668·42-s − 0.344·43-s + 1.20·46-s − 1.87·47-s + 1.00·48-s − 0.558·49-s − 0.994·54-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 500 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(4.815052428\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.815052428\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 3.01T + 9T^{2} \) |
| 7 | \( 1 - 4.64T + 49T^{2} \) |
| 11 | \( 1 - 121T^{2} \) |
| 13 | \( 1 - 169T^{2} \) |
| 17 | \( 1 - 289T^{2} \) |
| 19 | \( 1 - 361T^{2} \) |
| 23 | \( 1 - 27.7T + 529T^{2} \) |
| 29 | \( 1 + 57.8T + 841T^{2} \) |
| 31 | \( 1 - 961T^{2} \) |
| 37 | \( 1 - 1.36e3T^{2} \) |
| 41 | \( 1 - 70.1T + 1.68e3T^{2} \) |
| 43 | \( 1 + 14.7T + 1.84e3T^{2} \) |
| 47 | \( 1 + 88.0T + 2.20e3T^{2} \) |
| 53 | \( 1 - 2.80e3T^{2} \) |
| 59 | \( 1 - 3.48e3T^{2} \) |
| 61 | \( 1 - 110.T + 3.72e3T^{2} \) |
| 67 | \( 1 + 116T + 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 - 5.32e3T^{2} \) |
| 79 | \( 1 - 6.24e3T^{2} \) |
| 83 | \( 1 + 148.T + 6.88e3T^{2} \) |
| 89 | \( 1 - 51.7T + 7.92e3T^{2} \) |
| 97 | \( 1 - 9.40e3T^{2} \) |
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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
−11.14042864454322487946202386750, −9.814310026394094641994810923128, −8.814629709513882443064222303336, −7.88079056843301666550996549114, −7.16764070491405855031381652482, −5.88341326787590143791400789646, −4.92450431191085139685464495055, −3.80340086238676375279243311706, −2.84463167091546728681436237883, −1.72677802518089265333261828033,
1.72677802518089265333261828033, 2.84463167091546728681436237883, 3.80340086238676375279243311706, 4.92450431191085139685464495055, 5.88341326787590143791400789646, 7.16764070491405855031381652482, 7.88079056843301666550996549114, 8.814629709513882443064222303336, 9.814310026394094641994810923128, 11.14042864454322487946202386750
