| L(s) = 1 | + 2·2-s + 4·4-s + 7·5-s + 8·8-s + 14·10-s − 35·11-s − 66·13-s + 16·16-s + 59·17-s − 137·19-s + 28·20-s − 70·22-s + 7·23-s − 76·25-s − 132·26-s − 106·29-s − 75·31-s + 32·32-s + 118·34-s + 11·37-s − 274·38-s + 56·40-s − 498·41-s + 260·43-s − 140·44-s + 14·46-s − 171·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.626·5-s + 0.353·8-s + 0.442·10-s − 0.959·11-s − 1.40·13-s + 1/4·16-s + 0.841·17-s − 1.65·19-s + 0.313·20-s − 0.678·22-s + 0.0634·23-s − 0.607·25-s − 0.995·26-s − 0.678·29-s − 0.434·31-s + 0.176·32-s + 0.595·34-s + 0.0488·37-s − 1.16·38-s + 0.221·40-s − 1.89·41-s + 0.922·43-s − 0.479·44-s + 0.0448·46-s − 0.530·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 7 T + p^{3} T^{2} \) |
| 11 | \( 1 + 35 T + p^{3} T^{2} \) |
| 13 | \( 1 + 66 T + p^{3} T^{2} \) |
| 17 | \( 1 - 59 T + p^{3} T^{2} \) |
| 19 | \( 1 + 137 T + p^{3} T^{2} \) |
| 23 | \( 1 - 7 T + p^{3} T^{2} \) |
| 29 | \( 1 + 106 T + p^{3} T^{2} \) |
| 31 | \( 1 + 75 T + p^{3} T^{2} \) |
| 37 | \( 1 - 11 T + p^{3} T^{2} \) |
| 41 | \( 1 + 498 T + p^{3} T^{2} \) |
| 43 | \( 1 - 260 T + p^{3} T^{2} \) |
| 47 | \( 1 + 171 T + p^{3} T^{2} \) |
| 53 | \( 1 - 417 T + p^{3} T^{2} \) |
| 59 | \( 1 + 17 T + p^{3} T^{2} \) |
| 61 | \( 1 + 51 T + p^{3} T^{2} \) |
| 67 | \( 1 - 439 T + p^{3} T^{2} \) |
| 71 | \( 1 - 784 T + p^{3} T^{2} \) |
| 73 | \( 1 + 295 T + p^{3} T^{2} \) |
| 79 | \( 1 + 495 T + p^{3} T^{2} \) |
| 83 | \( 1 - 932 T + p^{3} T^{2} \) |
| 89 | \( 1 + 873 T + p^{3} T^{2} \) |
| 97 | \( 1 - 290 T + p^{3} T^{2} \) |
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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
−9.578025538396148571723150652048, −8.343154462127454312843443923003, −7.51250938607492249972758910901, −6.63683576660826544985377993427, −5.60072018701833752274212731263, −5.04844658809226600995849440569, −3.92731863407550956865679054011, −2.66387548266451829067739653876, −1.90833495296153328229970505997, 0,
1.90833495296153328229970505997, 2.66387548266451829067739653876, 3.92731863407550956865679054011, 5.04844658809226600995849440569, 5.60072018701833752274212731263, 6.63683576660826544985377993427, 7.51250938607492249972758910901, 8.343154462127454312843443923003, 9.578025538396148571723150652048
