L(s) = 1 | − 2-s + 4-s + 2·7-s − 8-s − 3·11-s + 5·13-s − 2·14-s + 16-s + 6·17-s − 4·19-s + 3·22-s − 3·23-s − 5·26-s + 2·28-s + 2·31-s − 32-s − 6·34-s + 11·37-s + 4·38-s + 6·41-s − 4·43-s − 3·44-s + 3·46-s + 3·47-s − 3·49-s + 5·52-s − 12·53-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.755·7-s − 0.353·8-s − 0.904·11-s + 1.38·13-s − 0.534·14-s + 1/4·16-s + 1.45·17-s − 0.917·19-s + 0.639·22-s − 0.625·23-s − 0.980·26-s + 0.377·28-s + 0.359·31-s − 0.176·32-s − 1.02·34-s + 1.80·37-s + 0.648·38-s + 0.937·41-s − 0.609·43-s − 0.452·44-s + 0.442·46-s + 0.437·47-s − 3/7·49-s + 0.693·52-s − 1.64·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.339697137\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.339697137\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 - 11 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−9.710360751401982168674084180337, −8.545596527007615457740236501561, −8.138387665244716020629763399489, −7.48719823111305954841245610417, −6.26379854421557824764190673304, −5.64923124129768131045952013240, −4.49990951754612574488554667858, −3.36928083217459561390498327423, −2.16765208032268725389293972277, −0.979666018302856449448839806449,
0.979666018302856449448839806449, 2.16765208032268725389293972277, 3.36928083217459561390498327423, 4.49990951754612574488554667858, 5.64923124129768131045952013240, 6.26379854421557824764190673304, 7.48719823111305954841245610417, 8.138387665244716020629763399489, 8.545596527007615457740236501561, 9.710360751401982168674084180337