L(s) = 1 | − 5-s + 4·7-s + 2·13-s + 6·17-s − 2·19-s + 6·23-s + 25-s − 6·29-s + 4·31-s − 4·35-s − 4·37-s − 43-s + 6·47-s + 9·49-s + 6·53-s + 12·59-s − 10·61-s − 2·65-s + 4·67-s − 4·73-s + 4·79-s − 6·83-s − 6·85-s + 6·89-s + 8·91-s + 2·95-s − 10·97-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 1.51·7-s + 0.554·13-s + 1.45·17-s − 0.458·19-s + 1.25·23-s + 1/5·25-s − 1.11·29-s + 0.718·31-s − 0.676·35-s − 0.657·37-s − 0.152·43-s + 0.875·47-s + 9/7·49-s + 0.824·53-s + 1.56·59-s − 1.28·61-s − 0.248·65-s + 0.488·67-s − 0.468·73-s + 0.450·79-s − 0.658·83-s − 0.650·85-s + 0.635·89-s + 0.838·91-s + 0.205·95-s − 1.01·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.192685671\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.192685671\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 43 | \( 1 + T \) |
good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−14.95563202909817, −14.72945911365083, −14.10136777861829, −13.60979570781375, −12.96930402175846, −12.35785542979178, −11.81661767462185, −11.41804922810108, −10.83228157900626, −10.48407806850573, −9.728702458626000, −8.992439227047922, −8.457545451378919, −8.108015160664720, −7.374919592434173, −7.112445875582425, −6.113242482002973, −5.478071799238423, −5.035813654049411, −4.351421045792665, −3.727554786832462, −3.063300569807305, −2.144868518699847, −1.380849532699156, −0.7536486302293520,
0.7536486302293520, 1.380849532699156, 2.144868518699847, 3.063300569807305, 3.727554786832462, 4.351421045792665, 5.035813654049411, 5.478071799238423, 6.113242482002973, 7.112445875582425, 7.374919592434173, 8.108015160664720, 8.457545451378919, 8.992439227047922, 9.728702458626000, 10.48407806850573, 10.83228157900626, 11.41804922810108, 11.81661767462185, 12.35785542979178, 12.96930402175846, 13.60979570781375, 14.10136777861829, 14.72945911365083, 14.95563202909817