L(s) = 1 | + 2·3-s − 3.46i·5-s + (2 + 1.73i)7-s + 9-s − 3.46i·11-s + 3.46i·13-s − 6.92i·15-s + 2·19-s + (4 + 3.46i)21-s − 3.46i·23-s − 6.99·25-s − 4·27-s − 6·29-s + 8·31-s − 6.92i·33-s + ⋯ |
L(s) = 1 | + 1.15·3-s − 1.54i·5-s + (0.755 + 0.654i)7-s + 0.333·9-s − 1.04i·11-s + 0.960i·13-s − 1.78i·15-s + 0.458·19-s + (0.872 + 0.755i)21-s − 0.722i·23-s − 1.39·25-s − 0.769·27-s − 1.11·29-s + 1.43·31-s − 1.20i·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 + 0.654i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.755 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.95274 - 0.728030i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.95274 - 0.728030i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (-2 - 1.73i)T \) |
good | 3 | \( 1 - 2T + 3T^{2} \) |
| 5 | \( 1 + 3.46iT - 5T^{2} \) |
| 11 | \( 1 + 3.46iT - 11T^{2} \) |
| 13 | \( 1 - 3.46iT - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 + 3.46iT - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 - 6.92iT - 41T^{2} \) |
| 43 | \( 1 - 10.3iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 - 3.46iT - 61T^{2} \) |
| 67 | \( 1 + 3.46iT - 67T^{2} \) |
| 71 | \( 1 - 3.46iT - 71T^{2} \) |
| 73 | \( 1 + 6.92iT - 73T^{2} \) |
| 79 | \( 1 - 3.46iT - 79T^{2} \) |
| 83 | \( 1 + 6T + 83T^{2} \) |
| 89 | \( 1 - 6.92iT - 89T^{2} \) |
| 97 | \( 1 + 13.8iT - 97T^{2} \) |
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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
−11.25706804491268867726777168842, −9.588953830211663260248387217318, −9.036128236832154519105890323137, −8.345910353654609925558497678554, −7.907070679167903386448456167174, −6.16478904905507191535804434872, −5.08387460991897900681181111709, −4.17067380789998926272961730796, −2.73617141151098625327292214363, −1.40351027887794056703929038807,
2.05462471309512005043083227882, 3.08433232669730184582890431382, 3.99748740106848241050424196853, 5.50244752816487965820653538501, 6.98708042166248761297055838228, 7.52496880825596917724798965123, 8.247623890924368214748670444184, 9.550059819250600182882265266760, 10.26292381518721065546968140955, 11.01287155568625878927977115204