| L(s) = 1 | + 5-s − 7-s − 6·13-s − 7·17-s + 8·19-s + 2·23-s − 4·25-s + 4·29-s + 2·31-s − 35-s + 9·37-s + 11·41-s + 7·43-s + 11·47-s + 49-s + 10·53-s − 9·59-s − 8·61-s − 6·65-s + 12·71-s − 14·73-s − 79-s + 9·83-s − 7·85-s + 14·89-s + 6·91-s + 8·95-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.377·7-s − 1.66·13-s − 1.69·17-s + 1.83·19-s + 0.417·23-s − 4/5·25-s + 0.742·29-s + 0.359·31-s − 0.169·35-s + 1.47·37-s + 1.71·41-s + 1.06·43-s + 1.60·47-s + 1/7·49-s + 1.37·53-s − 1.17·59-s − 1.02·61-s − 0.744·65-s + 1.42·71-s − 1.63·73-s − 0.112·79-s + 0.987·83-s − 0.759·85-s + 1.48·89-s + 0.628·91-s + 0.820·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.674440713\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.674440713\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 9 T + p T^{2} \) |
| 41 | \( 1 - 11 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 - 11 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 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
−9.110652078261170005377686510847, −7.66817430829402458063034646089, −7.40283459691100199015266370148, −6.41749827608374257598301306240, −5.71363303789453692926686983362, −4.82657381998998401025387317678, −4.14644619837587198466266087226, −2.78796365776706983964755303074, −2.33039492934956124296714217553, −0.77218417919464492872778356539,
0.77218417919464492872778356539, 2.33039492934956124296714217553, 2.78796365776706983964755303074, 4.14644619837587198466266087226, 4.82657381998998401025387317678, 5.71363303789453692926686983362, 6.41749827608374257598301306240, 7.40283459691100199015266370148, 7.66817430829402458063034646089, 9.110652078261170005377686510847
