L(s) = 1 | + 3.09·5-s − i·7-s + 0.646i·11-s + 4.37i·13-s + 0.295i·17-s + 5.29·19-s + 3.94·23-s + 4.58·25-s − 5.03·29-s − 9.16i·31-s − 3.09i·35-s + 2.55i·37-s + 10.4i·41-s + 1.63·43-s + 5.65·47-s + ⋯ |
L(s) = 1 | + 1.38·5-s − 0.377i·7-s + 0.194i·11-s + 1.21i·13-s + 0.0715i·17-s + 1.21·19-s + 0.823·23-s + 0.916·25-s − 0.934·29-s − 1.64i·31-s − 0.523i·35-s + 0.419i·37-s + 1.62i·41-s + 0.249·43-s + 0.825·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.938 - 0.346i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.938 - 0.346i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.672892561\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.672892561\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + iT \) |
good | 5 | \( 1 - 3.09T + 5T^{2} \) |
| 11 | \( 1 - 0.646iT - 11T^{2} \) |
| 13 | \( 1 - 4.37iT - 13T^{2} \) |
| 17 | \( 1 - 0.295iT - 17T^{2} \) |
| 19 | \( 1 - 5.29T + 19T^{2} \) |
| 23 | \( 1 - 3.94T + 23T^{2} \) |
| 29 | \( 1 + 5.03T + 29T^{2} \) |
| 31 | \( 1 + 9.16iT - 31T^{2} \) |
| 37 | \( 1 - 2.55iT - 37T^{2} \) |
| 41 | \( 1 - 10.4iT - 41T^{2} \) |
| 43 | \( 1 - 1.63T + 43T^{2} \) |
| 47 | \( 1 - 5.65T + 47T^{2} \) |
| 53 | \( 1 - 8.64T + 53T^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 - 6.92iT - 61T^{2} \) |
| 67 | \( 1 - 2.55T + 67T^{2} \) |
| 71 | \( 1 + 1.11T + 71T^{2} \) |
| 73 | \( 1 + 9.58T + 73T^{2} \) |
| 79 | \( 1 - 16.7iT - 79T^{2} \) |
| 83 | \( 1 + 8.50iT - 83T^{2} \) |
| 89 | \( 1 + 10.4iT - 89T^{2} \) |
| 97 | \( 1 + 2.41T + 97T^{2} \) |
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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
−8.680694140592326464874584575136, −7.54968449734941311873315414754, −7.05344619921147804855960833102, −6.17069412830034127847430221163, −5.64253195822907093500451820381, −4.77042010680623732791264680160, −3.97015570602627701201767208045, −2.83616340736826752750846085432, −1.98020939971037556012887490198, −1.11144066092977623967537942955,
0.880122482303472240044464635947, 1.94838550390134719203087879337, 2.84646165518699784038249342948, 3.56770362408708980483863939893, 5.00605107300911761079948680600, 5.50079204180040841871972895416, 5.92793574849634871239310232295, 6.97973369079446454514295356664, 7.55222064647235483342162987196, 8.625268176607400406730932685583