| L(s) = 1 | + 5-s + 4.10·7-s + 3.81·11-s + 5.81·13-s − 3.81·17-s − 1.81·19-s − 2.10·23-s + 25-s + 7.20·29-s − 1.81·31-s + 4.10·35-s − 6.01·37-s − 11.0·41-s + 5.81·43-s − 11.9·47-s + 9.81·49-s − 4.20·53-s + 3.81·55-s + 4.20·59-s − 3.01·61-s + 5.81·65-s + 3.71·67-s + 2.01·71-s + 8·73-s + 15.6·77-s + 2·79-s + 3.89·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.54·7-s + 1.15·11-s + 1.61·13-s − 0.925·17-s − 0.416·19-s − 0.438·23-s + 0.200·25-s + 1.33·29-s − 0.326·31-s + 0.693·35-s − 0.989·37-s − 1.72·41-s + 0.887·43-s − 1.73·47-s + 1.40·49-s − 0.577·53-s + 0.514·55-s + 0.547·59-s − 0.386·61-s + 0.721·65-s + 0.453·67-s + 0.239·71-s + 0.936·73-s + 1.78·77-s + 0.225·79-s + 0.427·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.488125449\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.488125449\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| good | 7 | \( 1 - 4.10T + 7T^{2} \) |
| 11 | \( 1 - 3.81T + 11T^{2} \) |
| 13 | \( 1 - 5.81T + 13T^{2} \) |
| 17 | \( 1 + 3.81T + 17T^{2} \) |
| 19 | \( 1 + 1.81T + 19T^{2} \) |
| 23 | \( 1 + 2.10T + 23T^{2} \) |
| 29 | \( 1 - 7.20T + 29T^{2} \) |
| 31 | \( 1 + 1.81T + 31T^{2} \) |
| 37 | \( 1 + 6.01T + 37T^{2} \) |
| 41 | \( 1 + 11.0T + 41T^{2} \) |
| 43 | \( 1 - 5.81T + 43T^{2} \) |
| 47 | \( 1 + 11.9T + 47T^{2} \) |
| 53 | \( 1 + 4.20T + 53T^{2} \) |
| 59 | \( 1 - 4.20T + 59T^{2} \) |
| 61 | \( 1 + 3.01T + 61T^{2} \) |
| 67 | \( 1 - 3.71T + 67T^{2} \) |
| 71 | \( 1 - 2.01T + 71T^{2} \) |
| 73 | \( 1 - 8T + 73T^{2} \) |
| 79 | \( 1 - 2T + 79T^{2} \) |
| 83 | \( 1 - 3.89T + 83T^{2} \) |
| 89 | \( 1 - 3T + 89T^{2} \) |
| 97 | \( 1 - 12.2T + 97T^{2} \) |
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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
−9.184336333505735567015642724355, −8.533170886892443980642322999345, −8.113383685935143694078510847133, −6.75992906393443556553762598542, −6.30875251118848548325784721591, −5.20800141788292909975440773623, −4.41931361589972766551397910478, −3.54797779988352701305034183370, −1.98715118664593145628317671147, −1.29617033210146696028229505400,
1.29617033210146696028229505400, 1.98715118664593145628317671147, 3.54797779988352701305034183370, 4.41931361589972766551397910478, 5.20800141788292909975440773623, 6.30875251118848548325784721591, 6.75992906393443556553762598542, 8.113383685935143694078510847133, 8.533170886892443980642322999345, 9.184336333505735567015642724355
