| L(s) = 1 | + 2·5-s − 7-s + 2·13-s − 6·17-s + 4·19-s + 4·23-s − 25-s + 6·29-s − 8·31-s − 2·35-s + 10·37-s + 10·41-s − 12·43-s + 8·47-s + 49-s + 6·53-s + 4·59-s + 10·61-s + 4·65-s − 12·67-s − 4·71-s + 2·73-s + 8·79-s + 4·83-s − 12·85-s − 6·89-s − 2·91-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.377·7-s + 0.554·13-s − 1.45·17-s + 0.917·19-s + 0.834·23-s − 1/5·25-s + 1.11·29-s − 1.43·31-s − 0.338·35-s + 1.64·37-s + 1.56·41-s − 1.82·43-s + 1.16·47-s + 1/7·49-s + 0.824·53-s + 0.520·59-s + 1.28·61-s + 0.496·65-s − 1.46·67-s − 0.474·71-s + 0.234·73-s + 0.900·79-s + 0.439·83-s − 1.30·85-s − 0.635·89-s − 0.209·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.219340409\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.219340409\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 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
−8.673879217932802792030912303788, −7.62235543862586666676262429236, −6.89231783790972432706288285438, −6.19687045589458139195079739208, −5.59977132734310801692380728532, −4.72543040100652910473023380683, −3.83703072789279610872308264668, −2.83015039799449329595938699080, −2.04489718779732369037105349689, −0.865532158136292932287895770899,
0.865532158136292932287895770899, 2.04489718779732369037105349689, 2.83015039799449329595938699080, 3.83703072789279610872308264668, 4.72543040100652910473023380683, 5.59977132734310801692380728532, 6.19687045589458139195079739208, 6.89231783790972432706288285438, 7.62235543862586666676262429236, 8.673879217932802792030912303788
