| L(s) = 1 | + 2.48·2-s + 4.17·4-s − 3.35·5-s + 7-s + 5.41·8-s − 8.33·10-s − 1.56·11-s + 0.560·13-s + 2.48·14-s + 5.10·16-s + 1.90·17-s + 3.15·19-s − 14.0·20-s − 3.88·22-s − 7.59·23-s + 6.23·25-s + 1.39·26-s + 4.17·28-s − 7.34·29-s − 3.72·31-s + 1.86·32-s + 4.73·34-s − 3.35·35-s − 4.98·37-s + 7.83·38-s − 18.1·40-s − 6.64·41-s + ⋯ |
| L(s) = 1 | + 1.75·2-s + 2.08·4-s − 1.49·5-s + 0.377·7-s + 1.91·8-s − 2.63·10-s − 0.471·11-s + 0.155·13-s + 0.664·14-s + 1.27·16-s + 0.462·17-s + 0.723·19-s − 3.13·20-s − 0.828·22-s − 1.58·23-s + 1.24·25-s + 0.273·26-s + 0.789·28-s − 1.36·29-s − 0.669·31-s + 0.329·32-s + 0.812·34-s − 0.566·35-s − 0.819·37-s + 1.27·38-s − 2.87·40-s − 1.03·41-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 | \( 1 - 2.48T + 2T^{2} \) |
| 5 | \( 1 + 3.35T + 5T^{2} \) |
| 11 | \( 1 + 1.56T + 11T^{2} \) |
| 13 | \( 1 - 0.560T + 13T^{2} \) |
| 17 | \( 1 - 1.90T + 17T^{2} \) |
| 19 | \( 1 - 3.15T + 19T^{2} \) |
| 23 | \( 1 + 7.59T + 23T^{2} \) |
| 29 | \( 1 + 7.34T + 29T^{2} \) |
| 31 | \( 1 + 3.72T + 31T^{2} \) |
| 37 | \( 1 + 4.98T + 37T^{2} \) |
| 41 | \( 1 + 6.64T + 41T^{2} \) |
| 43 | \( 1 + 1.33T + 43T^{2} \) |
| 47 | \( 1 - 2.63T + 47T^{2} \) |
| 53 | \( 1 - 2.87T + 53T^{2} \) |
| 59 | \( 1 - 11.5T + 59T^{2} \) |
| 61 | \( 1 - 0.252T + 61T^{2} \) |
| 67 | \( 1 + 1.89T + 67T^{2} \) |
| 71 | \( 1 + 7.84T + 71T^{2} \) |
| 73 | \( 1 - 10.3T + 73T^{2} \) |
| 79 | \( 1 + 1.28T + 79T^{2} \) |
| 83 | \( 1 + 9.61T + 83T^{2} \) |
| 89 | \( 1 - 1.02T + 89T^{2} \) |
| 97 | \( 1 + 0.963T + 97T^{2} \) |
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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
−7.30485900668946070561361985309, −6.83023445514852380593885525938, −5.69072633443093055330286625762, −5.39254375791882559068715974053, −4.55463289768325695036867491331, −3.73854755504553913611596036376, −3.64794206052276469751272828783, −2.60522914494694181685249160760, −1.62503100978663418218297203270, 0,
1.62503100978663418218297203270, 2.60522914494694181685249160760, 3.64794206052276469751272828783, 3.73854755504553913611596036376, 4.55463289768325695036867491331, 5.39254375791882559068715974053, 5.69072633443093055330286625762, 6.83023445514852380593885525938, 7.30485900668946070561361985309
