| L(s) = 1 | + 2·3-s − 7-s + 9-s − 5·11-s − 13-s + 4·17-s + 7·19-s − 2·21-s + 23-s − 4·27-s + 5·29-s + 2·31-s − 10·33-s + 2·37-s − 2·39-s + 11·41-s − 43-s + 8·47-s − 6·49-s + 8·51-s + 14·57-s − 14·59-s + 10·61-s − 63-s + 8·67-s + 2·69-s − 10·71-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.377·7-s + 1/3·9-s − 1.50·11-s − 0.277·13-s + 0.970·17-s + 1.60·19-s − 0.436·21-s + 0.208·23-s − 0.769·27-s + 0.928·29-s + 0.359·31-s − 1.74·33-s + 0.328·37-s − 0.320·39-s + 1.71·41-s − 0.152·43-s + 1.16·47-s − 6/7·49-s + 1.12·51-s + 1.85·57-s − 1.82·59-s + 1.28·61-s − 0.125·63-s + 0.977·67-s + 0.240·69-s − 1.18·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.609397357\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.609397357\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 11 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 - 7 T + p T^{2} \) |
| 83 | \( 1 - 15 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−8.119890266233448764673073286953, −7.75933913337492956708543983542, −7.23007117008369367255787736573, −6.03994603636818306950512062212, −5.37800953098479694736026160503, −4.58097637085452208340192660052, −3.38235747826954915221899397227, −2.97043737907793024466586140959, −2.26392041373491498443255092067, −0.840764340974826730158997134651,
0.840764340974826730158997134651, 2.26392041373491498443255092067, 2.97043737907793024466586140959, 3.38235747826954915221899397227, 4.58097637085452208340192660052, 5.37800953098479694736026160503, 6.03994603636818306950512062212, 7.23007117008369367255787736573, 7.75933913337492956708543983542, 8.119890266233448764673073286953
