L(s) = 1 | + 2-s + 4-s + 7-s + 8-s − 3·11-s − 5·13-s + 14-s + 16-s + 4·17-s + 19-s − 3·22-s + 8·23-s − 5·26-s + 28-s − 10·29-s − 8·31-s + 32-s + 4·34-s + 4·37-s + 38-s − 9·41-s − 43-s − 3·44-s + 8·46-s − 13·47-s + 49-s − 5·52-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.377·7-s + 0.353·8-s − 0.904·11-s − 1.38·13-s + 0.267·14-s + 1/4·16-s + 0.970·17-s + 0.229·19-s − 0.639·22-s + 1.66·23-s − 0.980·26-s + 0.188·28-s − 1.85·29-s − 1.43·31-s + 0.176·32-s + 0.685·34-s + 0.657·37-s + 0.162·38-s − 1.40·41-s − 0.152·43-s − 0.452·44-s + 1.17·46-s − 1.89·47-s + 1/7·49-s − 0.693·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 13 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + 15 T + p T^{2} \) |
| 89 | \( 1 - 11 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−7.23020346724023295715235118343, −6.87780545880214748996879866598, −5.58629589844294514229000995054, −5.30631382946002431245555925664, −4.81352372083824645681716851393, −3.76890647058484597839516015603, −3.09515664551022474479065769615, −2.33335115239000221504526264039, −1.45018831225646045227457082136, 0,
1.45018831225646045227457082136, 2.33335115239000221504526264039, 3.09515664551022474479065769615, 3.76890647058484597839516015603, 4.81352372083824645681716851393, 5.30631382946002431245555925664, 5.58629589844294514229000995054, 6.87780545880214748996879866598, 7.23020346724023295715235118343