L(s) = 1 | + (1.40 + 0.148i)2-s + (1.95 + 0.418i)4-s − 1.05i·5-s − i·7-s + (2.68 + 0.880i)8-s + (0.157 − 1.48i)10-s + 0.870·11-s + 2.65·13-s + (0.148 − 1.40i)14-s + (3.64 + 1.63i)16-s + 1.50i·17-s − 3.94i·19-s + (0.442 − 2.06i)20-s + (1.22 + 0.129i)22-s − 3.92·23-s + ⋯ |
L(s) = 1 | + (0.994 + 0.105i)2-s + (0.977 + 0.209i)4-s − 0.472i·5-s − 0.377i·7-s + (0.950 + 0.311i)8-s + (0.0497 − 0.469i)10-s + 0.262·11-s + 0.735·13-s + (0.0398 − 0.375i)14-s + (0.912 + 0.409i)16-s + 0.364i·17-s − 0.904i·19-s + (0.0989 − 0.461i)20-s + (0.260 + 0.0276i)22-s − 0.819·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 + 0.209i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.977 + 0.209i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.00088 - 0.317858i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.00088 - 0.317858i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.40 - 0.148i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + iT \) |
good | 5 | \( 1 + 1.05iT - 5T^{2} \) |
| 11 | \( 1 - 0.870T + 11T^{2} \) |
| 13 | \( 1 - 2.65T + 13T^{2} \) |
| 17 | \( 1 - 1.50iT - 17T^{2} \) |
| 19 | \( 1 + 3.94iT - 19T^{2} \) |
| 23 | \( 1 + 3.92T + 23T^{2} \) |
| 29 | \( 1 + 0.285iT - 29T^{2} \) |
| 31 | \( 1 + 0.345iT - 31T^{2} \) |
| 37 | \( 1 - 3.07T + 37T^{2} \) |
| 41 | \( 1 + 7.74iT - 41T^{2} \) |
| 43 | \( 1 - 8.68iT - 43T^{2} \) |
| 47 | \( 1 + 7.32T + 47T^{2} \) |
| 53 | \( 1 - 14.4iT - 53T^{2} \) |
| 59 | \( 1 + 10.7T + 59T^{2} \) |
| 61 | \( 1 + 3.78T + 61T^{2} \) |
| 67 | \( 1 + 5.17iT - 67T^{2} \) |
| 71 | \( 1 + 7.45T + 71T^{2} \) |
| 73 | \( 1 - 5.01T + 73T^{2} \) |
| 79 | \( 1 - 6.95iT - 79T^{2} \) |
| 83 | \( 1 + 12.7T + 83T^{2} \) |
| 89 | \( 1 - 14.9iT - 89T^{2} \) |
| 97 | \( 1 - 8.02T + 97T^{2} \) |
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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
−10.65278308885841053482206386769, −9.467921318958982863523563902015, −8.461554752422521052613370839169, −7.59169486349541624550820022221, −6.60136874233261774250398239697, −5.86190615278698542914206970578, −4.74977813973289834534110065713, −4.03263301327852190186854466115, −2.88996841351829289691915475935, −1.39896023479149093730690223369,
1.67747419286988775628713075389, 2.95904837375464593296085699055, 3.83401810267461509540089328158, 4.94083618002219171120509578233, 5.98064748383980058910117434329, 6.57090213606524817051684743368, 7.60975831813900090673344296472, 8.549879349289413872598766829778, 9.784430063344554991744127636763, 10.53408495353683460229976570646