L(s) = 1 | + 2.41i·3-s − 2.25i·5-s + (2.16 + 1.52i)7-s − 2.82·9-s + (0.870 − 3.20i)11-s + 13-s + 5.44·15-s + 1.61·17-s − 2.82·19-s + (−3.68 + 5.21i)21-s − 1.64·23-s − 0.0911·25-s + 0.429i·27-s + 0.879i·29-s + 0.464i·31-s + ⋯ |
L(s) = 1 | + 1.39i·3-s − 1.00i·5-s + (0.816 + 0.577i)7-s − 0.940·9-s + (0.262 − 0.964i)11-s + 0.277·13-s + 1.40·15-s + 0.390·17-s − 0.647·19-s + (−0.803 + 1.13i)21-s − 0.343·23-s − 0.0182·25-s + 0.0826i·27-s + 0.163i·29-s + 0.0833i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.771 - 0.636i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.771 - 0.636i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.204358689\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.204358689\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (-2.16 - 1.52i)T \) |
| 11 | \( 1 + (-0.870 + 3.20i)T \) |
| 13 | \( 1 - T \) |
good | 3 | \( 1 - 2.41iT - 3T^{2} \) |
| 5 | \( 1 + 2.25iT - 5T^{2} \) |
| 17 | \( 1 - 1.61T + 17T^{2} \) |
| 19 | \( 1 + 2.82T + 19T^{2} \) |
| 23 | \( 1 + 1.64T + 23T^{2} \) |
| 29 | \( 1 - 0.879iT - 29T^{2} \) |
| 31 | \( 1 - 0.464iT - 31T^{2} \) |
| 37 | \( 1 + 4.27T + 37T^{2} \) |
| 41 | \( 1 - 6.38T + 41T^{2} \) |
| 43 | \( 1 + 0.476iT - 43T^{2} \) |
| 47 | \( 1 + 11.5iT - 47T^{2} \) |
| 53 | \( 1 - 10.3T + 53T^{2} \) |
| 59 | \( 1 - 6.50iT - 59T^{2} \) |
| 61 | \( 1 - 8.25T + 61T^{2} \) |
| 67 | \( 1 - 3.29T + 67T^{2} \) |
| 71 | \( 1 + 4.29T + 71T^{2} \) |
| 73 | \( 1 - 16.0T + 73T^{2} \) |
| 79 | \( 1 - 6.07iT - 79T^{2} \) |
| 83 | \( 1 + 3.12T + 83T^{2} \) |
| 89 | \( 1 + 11.6iT - 89T^{2} \) |
| 97 | \( 1 + 10.9iT - 97T^{2} \) |
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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
−8.604019692238117747335862235005, −8.314861573629112999050346791043, −7.11132455626617925771742377278, −5.87431890083204072343758350398, −5.44074155757670638812836850379, −4.74694010506052642082115972784, −4.07350435085095459617900229671, −3.35253657705025738665313317783, −2.10000692752627268519630864459, −0.856330373970348941542361002574,
0.908385312187481654795750914889, 1.88268413198603244958852108843, 2.50416615297604723842455644716, 3.70942198009965298804788571311, 4.53129576555385718527373999021, 5.59639827397702297291308335603, 6.59131928043845924838088015227, 6.81281982483750629209850831684, 7.68886378202280529947827298559, 7.910976355949878272258354336481