L(s) = 1 | + (1.78 + 0.911i)2-s + (2.33 + 3.24i)4-s + (−1.00 − 1.00i)5-s + 10.0·7-s + (1.20 + 7.90i)8-s + (−0.875 − 2.71i)10-s + (−2.26 + 2.26i)11-s + (−6.88 + 6.88i)13-s + (17.8 + 9.13i)14-s + (−5.07 + 15.1i)16-s + 22.3·17-s + (−16.8 − 16.8i)19-s + (0.915 − 5.62i)20-s + (−6.09 + 1.96i)22-s − 33.2·23-s + ⋯ |
L(s) = 1 | + (0.890 + 0.455i)2-s + (0.584 + 0.811i)4-s + (−0.201 − 0.201i)5-s + 1.43·7-s + (0.150 + 0.988i)8-s + (−0.0875 − 0.271i)10-s + (−0.205 + 0.205i)11-s + (−0.529 + 0.529i)13-s + (1.27 + 0.652i)14-s + (−0.316 + 0.948i)16-s + 1.31·17-s + (−0.889 − 0.889i)19-s + (0.0457 − 0.281i)20-s + (−0.277 + 0.0894i)22-s − 1.44·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.655 - 0.754i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.655 - 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.24429 + 1.02334i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.24429 + 1.02334i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.78 - 0.911i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (1.00 + 1.00i)T + 25iT^{2} \) |
| 7 | \( 1 - 10.0T + 49T^{2} \) |
| 11 | \( 1 + (2.26 - 2.26i)T - 121iT^{2} \) |
| 13 | \( 1 + (6.88 - 6.88i)T - 169iT^{2} \) |
| 17 | \( 1 - 22.3T + 289T^{2} \) |
| 19 | \( 1 + (16.8 + 16.8i)T + 361iT^{2} \) |
| 23 | \( 1 + 33.2T + 529T^{2} \) |
| 29 | \( 1 + (-24.6 + 24.6i)T - 841iT^{2} \) |
| 31 | \( 1 + 41.3iT - 961T^{2} \) |
| 37 | \( 1 + (6.60 + 6.60i)T + 1.36e3iT^{2} \) |
| 41 | \( 1 - 47.1iT - 1.68e3T^{2} \) |
| 43 | \( 1 + (48.8 - 48.8i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + 45.6iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (25.1 + 25.1i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 + (6.23 - 6.23i)T - 3.48e3iT^{2} \) |
| 61 | \( 1 + (-35.9 + 35.9i)T - 3.72e3iT^{2} \) |
| 67 | \( 1 + (-10.2 - 10.2i)T + 4.48e3iT^{2} \) |
| 71 | \( 1 + 11.9T + 5.04e3T^{2} \) |
| 73 | \( 1 - 111. iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 4.46iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (10.1 + 10.1i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 - 21.9iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 107.T + 9.40e3T^{2} \) |
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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
−13.08538917883582524853210954768, −11.93083077749211224253090143741, −11.48226106428778770762916164191, −10.05575848726249632086470890771, −8.280581862329791243089597909348, −7.79379367935626944955094123774, −6.36156764574078866352997470779, −5.01057083568692735415638567865, −4.21623491613533889645173339489, −2.23407725220550549929032229678,
1.69491017288764090557004220601, 3.41864965395403729910714755149, 4.82558696578187842902816131349, 5.75582316581898929064531559277, 7.34871972003398701485825518481, 8.359781164880443851421843009511, 10.16625253058848583348422339637, 10.76856387561409349147339841093, 11.95737655115400107858833640132, 12.45368503921009534818124452911