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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.342 - 0.939i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3491633229\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3491633229\) |
\(L(\frac{3}{2})\) |
|
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
−8.563982896085881712097891215643, −7.76265375431372157059760807087, −7.08523660031560614436632805863, −6.51737063252432959883411440717, −6.08879829532493708298680643397, −5.23764139846194097033629779553, −3.78027983666053498986708185013, −2.95631139378980790574528687524, −2.40726959104765247079047751225, −1.23248335417649424801435905975,
0.10284258537618284400845437299, 1.56305117298412525565443427140, 2.94880241852237255792980804777, 3.85661494475180208957316480671, 4.39542863033988444584698884789, 5.01758433707169955302388037617, 5.72974746419061444825367205337, 6.86378011692102280342876500878, 7.40117286231867180411163381687, 8.617722517185997912059977733181