| L(s) = 1 | − i·3-s − 9-s − 3.46i·11-s + (3.46 − i)13-s − 6i·17-s − 6.92i·19-s + i·27-s + 6·29-s + 6.92i·31-s − 3.46·33-s + (−1 − 3.46i)39-s + 3.46i·41-s + 8i·43-s − 3.46·47-s − 7·49-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 0.333·9-s − 1.04i·11-s + (0.960 − 0.277i)13-s − 1.45i·17-s − 1.58i·19-s + 0.192i·27-s + 1.11·29-s + 1.24i·31-s − 0.603·33-s + (−0.160 − 0.554i)39-s + 0.541i·41-s + 1.21i·43-s − 0.505·47-s − 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.735 + 0.677i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.735 + 0.677i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.564693887\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.564693887\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 13 | \( 1 + (-3.46 + i)T \) |
| good | 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 3.46iT - 11T^{2} \) |
| 17 | \( 1 + 6iT - 17T^{2} \) |
| 19 | \( 1 + 6.92iT - 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 6.92iT - 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 - 3.46iT - 41T^{2} \) |
| 43 | \( 1 - 8iT - 43T^{2} \) |
| 47 | \( 1 + 3.46T + 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 + 3.46iT - 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 + 13.8T + 67T^{2} \) |
| 71 | \( 1 + 10.3iT - 71T^{2} \) |
| 73 | \( 1 + 6.92T + 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 3.46T + 83T^{2} \) |
| 89 | \( 1 + 17.3iT - 89T^{2} \) |
| 97 | \( 1 + 6.92T + 97T^{2} \) |
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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
−8.269041162155436205672949522398, −7.42819582351023791580830954917, −6.66477566881889370227963790680, −6.15730015343935767605867360791, −5.18694071390655008379142006984, −4.54150016490284399445372877401, −3.08879456257274220730890434915, −2.91379075472169989356434571561, −1.37261442810453604633869060856, −0.47854109356097716539497021244,
1.41092564711936391122574837245, 2.29175132605864843648540662612, 3.69642084620374911562154471429, 3.96773122702289156447331810119, 4.93220220023821060361914448697, 5.87988490871039589729180518116, 6.35597828484616799314966665754, 7.33909548093529925790404889686, 8.245201591144148874609629373942, 8.586295353166945523403628278929
