| L(s) = 1 | − 3.32·3-s + 1.40·5-s − 1.72·7-s + 8.08·9-s + 5.31·11-s − 13-s − 4.67·15-s − 0.616·17-s − 1.20·19-s + 5.72·21-s − 2.14·23-s − 3.02·25-s − 16.9·27-s + 29-s + 6.60·31-s − 17.7·33-s − 2.41·35-s + 1.28·37-s + 3.32·39-s + 0.566·41-s − 11.7·43-s + 11.3·45-s + 0.897·47-s − 4.03·49-s + 2.05·51-s − 13.3·53-s + 7.47·55-s + ⋯ |
| L(s) = 1 | − 1.92·3-s + 0.628·5-s − 0.650·7-s + 2.69·9-s + 1.60·11-s − 0.277·13-s − 1.20·15-s − 0.149·17-s − 0.276·19-s + 1.25·21-s − 0.446·23-s − 0.605·25-s − 3.25·27-s + 0.185·29-s + 1.18·31-s − 3.08·33-s − 0.408·35-s + 0.210·37-s + 0.533·39-s + 0.0885·41-s − 1.79·43-s + 1.69·45-s + 0.130·47-s − 0.577·49-s + 0.287·51-s − 1.83·53-s + 1.00·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 13 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 + 3.32T + 3T^{2} \) |
| 5 | \( 1 - 1.40T + 5T^{2} \) |
| 7 | \( 1 + 1.72T + 7T^{2} \) |
| 11 | \( 1 - 5.31T + 11T^{2} \) |
| 17 | \( 1 + 0.616T + 17T^{2} \) |
| 19 | \( 1 + 1.20T + 19T^{2} \) |
| 23 | \( 1 + 2.14T + 23T^{2} \) |
| 31 | \( 1 - 6.60T + 31T^{2} \) |
| 37 | \( 1 - 1.28T + 37T^{2} \) |
| 41 | \( 1 - 0.566T + 41T^{2} \) |
| 43 | \( 1 + 11.7T + 43T^{2} \) |
| 47 | \( 1 - 0.897T + 47T^{2} \) |
| 53 | \( 1 + 13.3T + 53T^{2} \) |
| 59 | \( 1 + 2.58T + 59T^{2} \) |
| 61 | \( 1 - 13.4T + 61T^{2} \) |
| 67 | \( 1 - 2.69T + 67T^{2} \) |
| 71 | \( 1 - 3.66T + 71T^{2} \) |
| 73 | \( 1 + 16.6T + 73T^{2} \) |
| 79 | \( 1 + 9.29T + 79T^{2} \) |
| 83 | \( 1 - 14.2T + 83T^{2} \) |
| 89 | \( 1 + 14.6T + 89T^{2} \) |
| 97 | \( 1 - 7.63T + 97T^{2} \) |
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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
−7.35975076399210063883008229638, −6.56937828237201615806166669421, −6.35925458626717662230928055177, −5.81007571643008222911923135944, −4.88725302236335235563716599359, −4.32360933708610531158016328508, −3.44830815224350609195764778623, −1.94758072377765008519112943160, −1.14625604419890471004806105329, 0,
1.14625604419890471004806105329, 1.94758072377765008519112943160, 3.44830815224350609195764778623, 4.32360933708610531158016328508, 4.88725302236335235563716599359, 5.81007571643008222911923135944, 6.35925458626717662230928055177, 6.56937828237201615806166669421, 7.35975076399210063883008229638
