L(s) = 1 | + (−0.0844 + 1.72i)3-s + 5-s − 1.02i·7-s + (−2.98 − 0.292i)9-s + 5.03·11-s + 5.07i·13-s + (−0.0844 + 1.72i)15-s + 3.80i·17-s − 7.38·19-s + (1.76 + 0.0863i)21-s − 4.35i·23-s + 25-s + (0.758 − 5.14i)27-s + 3.16i·29-s + 4.85i·31-s + ⋯ |
L(s) = 1 | + (−0.0487 + 0.998i)3-s + 0.447·5-s − 0.386i·7-s + (−0.995 − 0.0974i)9-s + 1.51·11-s + 1.40i·13-s + (−0.0218 + 0.446i)15-s + 0.923i·17-s − 1.69·19-s + (0.386 + 0.0188i)21-s − 0.907i·23-s + 0.200·25-s + (0.145 − 0.989i)27-s + 0.588i·29-s + 0.872i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.795 - 0.606i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.795 - 0.606i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.585893036\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.585893036\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.0844 - 1.72i)T \) |
| 5 | \( 1 - T \) |
| 67 | \( 1 + (-4.63 + 6.74i)T \) |
good | 7 | \( 1 + 1.02iT - 7T^{2} \) |
| 11 | \( 1 - 5.03T + 11T^{2} \) |
| 13 | \( 1 - 5.07iT - 13T^{2} \) |
| 17 | \( 1 - 3.80iT - 17T^{2} \) |
| 19 | \( 1 + 7.38T + 19T^{2} \) |
| 23 | \( 1 + 4.35iT - 23T^{2} \) |
| 29 | \( 1 - 3.16iT - 29T^{2} \) |
| 31 | \( 1 - 4.85iT - 31T^{2} \) |
| 37 | \( 1 - 6.59T + 37T^{2} \) |
| 41 | \( 1 + 0.222T + 41T^{2} \) |
| 43 | \( 1 + 2.43iT - 43T^{2} \) |
| 47 | \( 1 + 0.209iT - 47T^{2} \) |
| 53 | \( 1 + 14.0T + 53T^{2} \) |
| 59 | \( 1 - 12.5iT - 59T^{2} \) |
| 61 | \( 1 - 8.58iT - 61T^{2} \) |
| 71 | \( 1 + 6.87iT - 71T^{2} \) |
| 73 | \( 1 + 16.1T + 73T^{2} \) |
| 79 | \( 1 - 2.75iT - 79T^{2} \) |
| 83 | \( 1 - 15.6iT - 83T^{2} \) |
| 89 | \( 1 - 3.59iT - 89T^{2} \) |
| 97 | \( 1 + 0.794iT - 97T^{2} \) |
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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
−8.878837666866245288753065312747, −8.373481494591525609318541320256, −7.02837269204447987668132984629, −6.38605535943368749436895587046, −5.97428865362456719031004860876, −4.61730084439503163729021559631, −4.29198047645940315982704202239, −3.60979341586638026650100184388, −2.36737596675817420087588382749, −1.37472803198347482772392969849,
0.45544851265235649100193854924, 1.56163335195159531785093648316, 2.43650783969933850788292111652, 3.28421753157903705377819021667, 4.39004843611037366418737594198, 5.41441961168100386182876581877, 6.15640837217127090871633578113, 6.47856197890633156554984613160, 7.48647212905509210432808564275, 8.052358746580287935559263225682