| L(s) = 1 | + 1.00i·3-s − 1.31i·5-s + (−1.40 + 2.24i)7-s + 1.98·9-s + (−2.38 − 2.30i)11-s + 13-s + 1.32·15-s − 1.70·17-s + 7.19·19-s + (−2.25 − 1.41i)21-s − 3.72·23-s + 3.27·25-s + 5.01i·27-s − 7.88i·29-s + 5.82i·31-s + ⋯ |
| L(s) = 1 | + 0.581i·3-s − 0.587i·5-s + (−0.529 + 0.848i)7-s + 0.662·9-s + (−0.717 − 0.696i)11-s + 0.277·13-s + 0.341·15-s − 0.412·17-s + 1.65·19-s + (−0.492 − 0.307i)21-s − 0.777·23-s + 0.655·25-s + 0.965i·27-s − 1.46i·29-s + 1.04i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.239i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 - 0.239i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.801724669\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.801724669\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (1.40 - 2.24i)T \) |
| 11 | \( 1 + (2.38 + 2.30i)T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 - 1.00iT - 3T^{2} \) |
| 5 | \( 1 + 1.31iT - 5T^{2} \) |
| 17 | \( 1 + 1.70T + 17T^{2} \) |
| 19 | \( 1 - 7.19T + 19T^{2} \) |
| 23 | \( 1 + 3.72T + 23T^{2} \) |
| 29 | \( 1 + 7.88iT - 29T^{2} \) |
| 31 | \( 1 - 5.82iT - 31T^{2} \) |
| 37 | \( 1 + 8.84T + 37T^{2} \) |
| 41 | \( 1 - 7.03T + 41T^{2} \) |
| 43 | \( 1 + 7.07iT - 43T^{2} \) |
| 47 | \( 1 - 3.62iT - 47T^{2} \) |
| 53 | \( 1 - 6.34T + 53T^{2} \) |
| 59 | \( 1 + 1.65iT - 59T^{2} \) |
| 61 | \( 1 + 3.42T + 61T^{2} \) |
| 67 | \( 1 - 13.5T + 67T^{2} \) |
| 71 | \( 1 - 7.55T + 71T^{2} \) |
| 73 | \( 1 + 11.0T + 73T^{2} \) |
| 79 | \( 1 + 12.8iT - 79T^{2} \) |
| 83 | \( 1 - 15.2T + 83T^{2} \) |
| 89 | \( 1 - 10.7iT - 89T^{2} \) |
| 97 | \( 1 + 1.83iT - 97T^{2} \) |
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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
−8.650076252802682007621348164750, −7.85726946918252851673999210182, −7.02839182165407882050927871208, −6.09471537086649140359552582821, −5.37789480973186450501854829915, −4.87025034776559566156551713034, −3.81447627432765697810310544038, −3.13583976832126562248231262632, −2.07867395920731246158559200189, −0.74263314249050828096105757441,
0.811502224177113672490158830497, 1.88546146678353908175823468775, 2.92327468298270119794145256848, 3.72693240066456483067179350953, 4.60203182632704529567467029752, 5.50209559774651303116777649020, 6.48631537098708141704207725683, 7.11822076273578656165652537672, 7.40525740975414800402703631238, 8.166434669679150974984918057603
