| 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
| \(L(1)\) |
\(\approx\) |
\(9.484972494\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.484972494\) |
| \(L(\frac{3}{2})\) |
|
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
−7.28261661937334034487211081292, −6.83607001157554842947575385821, −6.14404271778757212963761193298, −5.69206239101514900519755268710, −5.21001660797597958829976342240, −4.35682714839795440951971789165, −3.49758199328330982358108085229, −2.91098170062477622393544001877, −1.99867040090942002618330155814, −1.34641723071215068750780706853,
1.34641723071215068750780706853, 1.99867040090942002618330155814, 2.91098170062477622393544001877, 3.49758199328330982358108085229, 4.35682714839795440951971789165, 5.21001660797597958829976342240, 5.69206239101514900519755268710, 6.14404271778757212963761193298, 6.83607001157554842947575385821, 7.28261661937334034487211081292
