| L(s) = 1 | + 5-s − 4·7-s + 6·11-s − 13-s + 2·19-s − 23-s + 25-s − 9·29-s + 5·31-s − 4·35-s + 2·37-s + 9·41-s − 4·43-s + 3·47-s + 9·49-s + 6·53-s + 6·55-s + 2·61-s − 65-s − 10·67-s + 3·71-s − 7·73-s − 24·77-s − 10·79-s + 12·83-s + 4·91-s + 2·95-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.51·7-s + 1.80·11-s − 0.277·13-s + 0.458·19-s − 0.208·23-s + 1/5·25-s − 1.67·29-s + 0.898·31-s − 0.676·35-s + 0.328·37-s + 1.40·41-s − 0.609·43-s + 0.437·47-s + 9/7·49-s + 0.824·53-s + 0.809·55-s + 0.256·61-s − 0.124·65-s − 1.22·67-s + 0.356·71-s − 0.819·73-s − 2.73·77-s − 1.12·79-s + 1.31·83-s + 0.419·91-s + 0.205·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.828241162\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.828241162\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−8.675195224509421244567010783131, −7.44458669533489548487414918137, −6.92397468338108808019310143609, −6.11387858034595661701643360874, −5.81664617436965714998858914525, −4.51281009230397575705652247627, −3.74499491504031266853477766636, −3.05859762092563524808570267988, −1.96419494679653881817999121224, −0.77418837928818037905531196406,
0.77418837928818037905531196406, 1.96419494679653881817999121224, 3.05859762092563524808570267988, 3.74499491504031266853477766636, 4.51281009230397575705652247627, 5.81664617436965714998858914525, 6.11387858034595661701643360874, 6.92397468338108808019310143609, 7.44458669533489548487414918137, 8.675195224509421244567010783131
