| L(s) = 1 | − 2.51·2-s + 4.32·4-s + 1.19·5-s − 7-s − 5.85·8-s − 2.99·10-s + 4.18·11-s + 1.43·13-s + 2.51·14-s + 6.07·16-s − 1.76·17-s − 4.92·19-s + 5.15·20-s − 10.5·22-s + 0.661·23-s − 3.58·25-s − 3.60·26-s − 4.32·28-s + 0.521·29-s + 3.67·31-s − 3.57·32-s + 4.43·34-s − 1.19·35-s + 1.30·37-s + 12.3·38-s − 6.97·40-s − 8.43·41-s + ⋯ |
| L(s) = 1 | − 1.77·2-s + 2.16·4-s + 0.532·5-s − 0.377·7-s − 2.07·8-s − 0.947·10-s + 1.26·11-s + 0.397·13-s + 0.672·14-s + 1.51·16-s − 0.427·17-s − 1.13·19-s + 1.15·20-s − 2.24·22-s + 0.137·23-s − 0.716·25-s − 0.707·26-s − 0.818·28-s + 0.0967·29-s + 0.659·31-s − 0.632·32-s + 0.760·34-s − 0.201·35-s + 0.214·37-s + 2.01·38-s − 1.10·40-s − 1.31·41-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.51T + 2T^{2} \) |
| 5 | \( 1 - 1.19T + 5T^{2} \) |
| 11 | \( 1 - 4.18T + 11T^{2} \) |
| 13 | \( 1 - 1.43T + 13T^{2} \) |
| 17 | \( 1 + 1.76T + 17T^{2} \) |
| 19 | \( 1 + 4.92T + 19T^{2} \) |
| 23 | \( 1 - 0.661T + 23T^{2} \) |
| 29 | \( 1 - 0.521T + 29T^{2} \) |
| 31 | \( 1 - 3.67T + 31T^{2} \) |
| 37 | \( 1 - 1.30T + 37T^{2} \) |
| 41 | \( 1 + 8.43T + 41T^{2} \) |
| 43 | \( 1 - 4.56T + 43T^{2} \) |
| 47 | \( 1 + 0.797T + 47T^{2} \) |
| 53 | \( 1 - 9.82T + 53T^{2} \) |
| 59 | \( 1 + 12.7T + 59T^{2} \) |
| 61 | \( 1 - 0.0608T + 61T^{2} \) |
| 67 | \( 1 + 7.96T + 67T^{2} \) |
| 71 | \( 1 + 6.48T + 71T^{2} \) |
| 73 | \( 1 + 9.61T + 73T^{2} \) |
| 79 | \( 1 - 15.3T + 79T^{2} \) |
| 83 | \( 1 + 5.82T + 83T^{2} \) |
| 89 | \( 1 + 12.1T + 89T^{2} \) |
| 97 | \( 1 - 0.0612T + 97T^{2} \) |
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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
−7.56844747100166638389826166045, −6.92876953819428976535881581315, −6.24847163294540894956143432147, −6.01093154279003753297758215981, −4.60524878897572354738533150633, −3.73315001953263357281677912169, −2.68791513094229719064043369224, −1.87715281365477109109653670399, −1.18252691627433909880581280637, 0,
1.18252691627433909880581280637, 1.87715281365477109109653670399, 2.68791513094229719064043369224, 3.73315001953263357281677912169, 4.60524878897572354738533150633, 6.01093154279003753297758215981, 6.24847163294540894956143432147, 6.92876953819428976535881581315, 7.56844747100166638389826166045
