L(s) = 1 | − 1.59·2-s + 0.530·4-s + 1.30·5-s + 7-s + 2.33·8-s − 2.07·10-s − 0.521·11-s − 5.72·13-s − 1.59·14-s − 4.77·16-s + 0.651·17-s + 4.34·19-s + 0.691·20-s + 0.829·22-s − 0.286·23-s − 3.29·25-s + 9.09·26-s + 0.530·28-s + 9.23·29-s − 3.19·31-s + 2.92·32-s − 1.03·34-s + 1.30·35-s − 6.41·37-s − 6.91·38-s + 3.04·40-s − 7.05·41-s + ⋯ |
L(s) = 1 | − 1.12·2-s + 0.265·4-s + 0.583·5-s + 0.377·7-s + 0.826·8-s − 0.655·10-s − 0.157·11-s − 1.58·13-s − 0.425·14-s − 1.19·16-s + 0.158·17-s + 0.997·19-s + 0.154·20-s + 0.176·22-s − 0.0597·23-s − 0.659·25-s + 1.78·26-s + 0.100·28-s + 1.71·29-s − 0.574·31-s + 0.517·32-s − 0.177·34-s + 0.220·35-s − 1.05·37-s − 1.12·38-s + 0.481·40-s − 1.10·41-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
\(L(1)\) |
\(\approx\) |
\(0.9480930045\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9480930045\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.59T + 2T^{2} \) |
| 5 | \( 1 - 1.30T + 5T^{2} \) |
| 11 | \( 1 + 0.521T + 11T^{2} \) |
| 13 | \( 1 + 5.72T + 13T^{2} \) |
| 17 | \( 1 - 0.651T + 17T^{2} \) |
| 19 | \( 1 - 4.34T + 19T^{2} \) |
| 23 | \( 1 + 0.286T + 23T^{2} \) |
| 29 | \( 1 - 9.23T + 29T^{2} \) |
| 31 | \( 1 + 3.19T + 31T^{2} \) |
| 37 | \( 1 + 6.41T + 37T^{2} \) |
| 41 | \( 1 + 7.05T + 41T^{2} \) |
| 43 | \( 1 - 6.08T + 43T^{2} \) |
| 47 | \( 1 + 4.36T + 47T^{2} \) |
| 53 | \( 1 - 4.71T + 53T^{2} \) |
| 59 | \( 1 + 6.14T + 59T^{2} \) |
| 61 | \( 1 + 0.825T + 61T^{2} \) |
| 67 | \( 1 - 5.94T + 67T^{2} \) |
| 71 | \( 1 - 11.6T + 71T^{2} \) |
| 73 | \( 1 + 12.7T + 73T^{2} \) |
| 79 | \( 1 - 2.01T + 79T^{2} \) |
| 83 | \( 1 + 11.6T + 83T^{2} \) |
| 89 | \( 1 - 7.39T + 89T^{2} \) |
| 97 | \( 1 - 13.1T + 97T^{2} \) |
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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
−7.87650108196704170918218623547, −7.33989663957890191893532644320, −6.77477880440903320740579163590, −5.71289221599201635665814667267, −5.01457718642875991988241633669, −4.53221887458722707324403401936, −3.32290189438326355087814749758, −2.33129350398377225524884936292, −1.65313976256958056361932662234, −0.58235554120775856657681116204,
0.58235554120775856657681116204, 1.65313976256958056361932662234, 2.33129350398377225524884936292, 3.32290189438326355087814749758, 4.53221887458722707324403401936, 5.01457718642875991988241633669, 5.71289221599201635665814667267, 6.77477880440903320740579163590, 7.33989663957890191893532644320, 7.87650108196704170918218623547