L(s) = 1 | − i·2-s − 4-s − i·5-s − 1.30·7-s + i·8-s − 10-s − 0.690·11-s − 2.69i·13-s + 1.30i·14-s + 16-s − 6.90i·17-s + 2.69i·19-s + i·20-s + 0.690i·22-s − 8.90i·23-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.5·4-s − 0.447i·5-s − 0.495·7-s + 0.353i·8-s − 0.316·10-s − 0.208·11-s − 0.746i·13-s + 0.350i·14-s + 0.250·16-s − 1.67i·17-s + 0.617i·19-s + 0.223i·20-s + 0.147i·22-s − 1.85i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.675 - 0.737i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.675 - 0.737i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5262145061\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5262145061\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 + iT \) |
| 37 | \( 1 + (-4.10 - 4.48i)T \) |
good | 7 | \( 1 + 1.30T + 7T^{2} \) |
| 11 | \( 1 + 0.690T + 11T^{2} \) |
| 13 | \( 1 + 2.69iT - 13T^{2} \) |
| 17 | \( 1 + 6.90iT - 17T^{2} \) |
| 19 | \( 1 - 2.69iT - 19T^{2} \) |
| 23 | \( 1 + 8.90iT - 23T^{2} \) |
| 29 | \( 1 - 2iT - 29T^{2} \) |
| 31 | \( 1 - 9.59iT - 31T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 8.21iT - 43T^{2} \) |
| 47 | \( 1 - 4.61T + 47T^{2} \) |
| 53 | \( 1 + 0.690T + 53T^{2} \) |
| 59 | \( 1 - 6.21iT - 59T^{2} \) |
| 61 | \( 1 + 2iT - 61T^{2} \) |
| 67 | \( 1 + 1.23T + 67T^{2} \) |
| 71 | \( 1 + 13.8T + 71T^{2} \) |
| 73 | \( 1 + 12.2T + 73T^{2} \) |
| 79 | \( 1 + 1.59iT - 79T^{2} \) |
| 83 | \( 1 + 10.2T + 83T^{2} \) |
| 89 | \( 1 + 5.92iT - 89T^{2} \) |
| 97 | \( 1 - 3.38iT - 97T^{2} \) |
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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
−8.417184315549201249539245463533, −7.47311733751355557569427970530, −6.69595871306119300560412062937, −5.69700045044607079584507267548, −4.97162906455190267155951724630, −4.27466768848111523994091600264, −3.11481221499821211685596745131, −2.63307335500337638059311192294, −1.23588283155390227869120259849, −0.16903558315963790716294049669,
1.57784748886960220896653656684, 2.77953127010587209090532147899, 3.84191628996425138598798106390, 4.37703866835810480092881128896, 5.67622544107701968304139562046, 6.05048519024268534738842772657, 6.84410541771061295283992573817, 7.61151200920662697442673833122, 8.127223085551192063591751930413, 9.194975301818258551153369681973