L(s) = 1 | − 2.47·2-s + 4.12·4-s − 3.91·5-s − 7-s − 5.25·8-s + 9.69·10-s − 2.92·11-s + 1.07·13-s + 2.47·14-s + 4.76·16-s − 1.80·17-s + 1.65·19-s − 16.1·20-s + 7.22·22-s − 2.10·23-s + 10.3·25-s − 2.66·26-s − 4.12·28-s + 4.17·29-s − 1.66·31-s − 1.26·32-s + 4.46·34-s + 3.91·35-s − 1.01·37-s − 4.10·38-s + 20.5·40-s + 8.34·41-s + ⋯ |
L(s) = 1 | − 1.74·2-s + 2.06·4-s − 1.75·5-s − 0.377·7-s − 1.85·8-s + 3.06·10-s − 0.880·11-s + 0.298·13-s + 0.661·14-s + 1.19·16-s − 0.437·17-s + 0.380·19-s − 3.61·20-s + 1.54·22-s − 0.439·23-s + 2.06·25-s − 0.522·26-s − 0.779·28-s + 0.775·29-s − 0.299·31-s − 0.224·32-s + 0.765·34-s + 0.662·35-s − 0.166·37-s − 0.666·38-s + 3.25·40-s + 1.30·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1667096860\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1667096860\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 127 | \( 1 - T \) |
good | 2 | \( 1 + 2.47T + 2T^{2} \) |
| 5 | \( 1 + 3.91T + 5T^{2} \) |
| 11 | \( 1 + 2.92T + 11T^{2} \) |
| 13 | \( 1 - 1.07T + 13T^{2} \) |
| 17 | \( 1 + 1.80T + 17T^{2} \) |
| 19 | \( 1 - 1.65T + 19T^{2} \) |
| 23 | \( 1 + 2.10T + 23T^{2} \) |
| 29 | \( 1 - 4.17T + 29T^{2} \) |
| 31 | \( 1 + 1.66T + 31T^{2} \) |
| 37 | \( 1 + 1.01T + 37T^{2} \) |
| 41 | \( 1 - 8.34T + 41T^{2} \) |
| 43 | \( 1 + 2.95T + 43T^{2} \) |
| 47 | \( 1 + 1.91T + 47T^{2} \) |
| 53 | \( 1 + 10.9T + 53T^{2} \) |
| 59 | \( 1 + 7.84T + 59T^{2} \) |
| 61 | \( 1 + 10.6T + 61T^{2} \) |
| 67 | \( 1 + 8.08T + 67T^{2} \) |
| 71 | \( 1 - 10.7T + 71T^{2} \) |
| 73 | \( 1 - 0.791T + 73T^{2} \) |
| 79 | \( 1 + 0.860T + 79T^{2} \) |
| 83 | \( 1 - 2.57T + 83T^{2} \) |
| 89 | \( 1 - 9.63T + 89T^{2} \) |
| 97 | \( 1 + 5.99T + 97T^{2} \) |
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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
−7.80470169811862236478435994143, −7.62873131843986709268345470166, −6.78001941737500739804753779068, −6.19181118916951742690779573120, −4.98687941037146743410581551260, −4.16332876111473943817697116363, −3.23394460652240516849540706170, −2.59901896579357399501498458360, −1.34591489373139669539949090506, −0.28057568247707568064464055249,
0.28057568247707568064464055249, 1.34591489373139669539949090506, 2.59901896579357399501498458360, 3.23394460652240516849540706170, 4.16332876111473943817697116363, 4.98687941037146743410581551260, 6.19181118916951742690779573120, 6.78001941737500739804753779068, 7.62873131843986709268345470166, 7.80470169811862236478435994143