| L(s) = 1 | + 2.25·2-s + 3-s + 3.07·4-s + 1.02·5-s + 2.25·6-s − 7-s + 2.43·8-s + 9-s + 2.31·10-s − 4.12·11-s + 3.07·12-s − 4.95·13-s − 2.25·14-s + 1.02·15-s − 0.678·16-s + 5.40·17-s + 2.25·18-s − 3.35·19-s + 3.15·20-s − 21-s − 9.29·22-s − 3.01·23-s + 2.43·24-s − 3.94·25-s − 11.1·26-s + 27-s − 3.07·28-s + ⋯ |
| L(s) = 1 | + 1.59·2-s + 0.577·3-s + 1.53·4-s + 0.458·5-s + 0.920·6-s − 0.377·7-s + 0.859·8-s + 0.333·9-s + 0.730·10-s − 1.24·11-s + 0.888·12-s − 1.37·13-s − 0.602·14-s + 0.264·15-s − 0.169·16-s + 1.31·17-s + 0.531·18-s − 0.768·19-s + 0.705·20-s − 0.218·21-s − 1.98·22-s − 0.629·23-s + 0.496·24-s − 0.789·25-s − 2.19·26-s + 0.192·27-s − 0.581·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 383 | \( 1 - T \) |
| good | 2 | \( 1 - 2.25T + 2T^{2} \) |
| 5 | \( 1 - 1.02T + 5T^{2} \) |
| 11 | \( 1 + 4.12T + 11T^{2} \) |
| 13 | \( 1 + 4.95T + 13T^{2} \) |
| 17 | \( 1 - 5.40T + 17T^{2} \) |
| 19 | \( 1 + 3.35T + 19T^{2} \) |
| 23 | \( 1 + 3.01T + 23T^{2} \) |
| 29 | \( 1 + 6.00T + 29T^{2} \) |
| 31 | \( 1 + 2.64T + 31T^{2} \) |
| 37 | \( 1 + 10.0T + 37T^{2} \) |
| 41 | \( 1 + 5.27T + 41T^{2} \) |
| 43 | \( 1 + 5.56T + 43T^{2} \) |
| 47 | \( 1 - 7.18T + 47T^{2} \) |
| 53 | \( 1 + 1.79T + 53T^{2} \) |
| 59 | \( 1 - 6.99T + 59T^{2} \) |
| 61 | \( 1 - 5.31T + 61T^{2} \) |
| 67 | \( 1 - 7.39T + 67T^{2} \) |
| 71 | \( 1 - 13.6T + 71T^{2} \) |
| 73 | \( 1 + 14.3T + 73T^{2} \) |
| 79 | \( 1 - 12.5T + 79T^{2} \) |
| 83 | \( 1 - 2.53T + 83T^{2} \) |
| 89 | \( 1 - 10.3T + 89T^{2} \) |
| 97 | \( 1 - 4.92T + 97T^{2} \) |
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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
−7.35600572425938580979586884766, −6.68872382155784141600556274341, −5.76864161462960763031268265944, −5.33104163030007924045677317581, −4.79398145750821498150346845971, −3.71224207915011807171553187402, −3.36557844887802468801907842008, −2.28649468252077605877372984862, −2.07408755504667097947931479573, 0,
2.07408755504667097947931479573, 2.28649468252077605877372984862, 3.36557844887802468801907842008, 3.71224207915011807171553187402, 4.79398145750821498150346845971, 5.33104163030007924045677317581, 5.76864161462960763031268265944, 6.68872382155784141600556274341, 7.35600572425938580979586884766
