| L(s) = 1 | + (−1 + 2i)5-s − i·7-s + 3·9-s − 4i·13-s − 4i·17-s + 4·19-s + 8i·23-s + (−3 − 4i)25-s + 2·29-s − 8·31-s + (2 + i)35-s − 8i·37-s + 6·41-s − 8i·43-s + (−3 + 6i)45-s + ⋯ |
| L(s) = 1 | + (−0.447 + 0.894i)5-s − 0.377i·7-s + 9-s − 1.10i·13-s − 0.970i·17-s + 0.917·19-s + 1.66i·23-s + (−0.600 − 0.800i)25-s + 0.371·29-s − 1.43·31-s + (0.338 + 0.169i)35-s − 1.31i·37-s + 0.937·41-s − 1.21i·43-s + (−0.447 + 0.894i)45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.685355674\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.685355674\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (1 - 2i)T \) |
| 7 | \( 1 + iT \) |
| good | 3 | \( 1 - 3T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 4iT - 13T^{2} \) |
| 17 | \( 1 + 4iT - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 - 8iT - 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + 8iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 8iT - 43T^{2} \) |
| 47 | \( 1 + 8iT - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 - 8iT - 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 + 4iT - 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 - 12iT - 97T^{2} \) |
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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
−9.137686707900680085913476179534, −7.85736555584818748408962285992, −7.36280658329113588010497697489, −7.01640001503473046161691473715, −5.76237766064378023577312360636, −5.06992073339014491621673593167, −3.80896328457004026173143164995, −3.38750467354747029544868907039, −2.16125677535040774435472701818, −0.70871702573805387919943549440,
1.09118947008733293021346928441, 2.05657384643093579597724417832, 3.48217703859000083218707859499, 4.41892627409401099006273881689, 4.83554825728377376280551939183, 6.01796098683231584210676222744, 6.75136866520401204692980621471, 7.67548951854078018943757900977, 8.314695529438921143449331470373, 9.153473697376734019700972831067
