| L(s) = 1 | + 4.62·2-s − 3·3-s + 13.4·4-s − 5.80·5-s − 13.8·6-s − 2.66·7-s + 24.9·8-s + 9·9-s − 26.8·10-s − 11·11-s − 40.2·12-s − 1.17·13-s − 12.3·14-s + 17.4·15-s + 8.37·16-s − 23.4·17-s + 41.6·18-s + 158.·19-s − 77.7·20-s + 7.99·21-s − 50.8·22-s + 121.·23-s − 74.9·24-s − 91.2·25-s − 5.43·26-s − 27·27-s − 35.7·28-s + ⋯ |
| L(s) = 1 | + 1.63·2-s − 0.577·3-s + 1.67·4-s − 0.519·5-s − 0.944·6-s − 0.143·7-s + 1.10·8-s + 0.333·9-s − 0.849·10-s − 0.301·11-s − 0.967·12-s − 0.0250·13-s − 0.235·14-s + 0.299·15-s + 0.130·16-s − 0.334·17-s + 0.545·18-s + 1.91·19-s − 0.869·20-s + 0.0830·21-s − 0.493·22-s + 1.10·23-s − 0.637·24-s − 0.730·25-s − 0.0409·26-s − 0.192·27-s − 0.241·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 11 | \( 1 + 11T \) |
| 61 | \( 1 - 61T \) |
| good | 2 | \( 1 - 4.62T + 8T^{2} \) |
| 5 | \( 1 + 5.80T + 125T^{2} \) |
| 7 | \( 1 + 2.66T + 343T^{2} \) |
| 13 | \( 1 + 1.17T + 2.19e3T^{2} \) |
| 17 | \( 1 + 23.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 158.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 121.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 32.3T + 2.43e4T^{2} \) |
| 31 | \( 1 - 54.0T + 2.97e4T^{2} \) |
| 37 | \( 1 + 374.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 254.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 360.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 576.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 138.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 707.T + 2.05e5T^{2} \) |
| 67 | \( 1 + 493.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 181.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 598.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.00e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 634.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.09e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 437.T + 9.12e5T^{2} \) |
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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
−8.143980170472895790419522408103, −7.16908094728150537899855587269, −6.74294969418301968956156051808, −5.65916525676598715645881358028, −5.18811576032154250413803027584, −4.44036041113242420017292655371, −3.49249430982653305042215872424, −2.87870591663049314811309896167, −1.47713689851859188354647147877, 0,
1.47713689851859188354647147877, 2.87870591663049314811309896167, 3.49249430982653305042215872424, 4.44036041113242420017292655371, 5.18811576032154250413803027584, 5.65916525676598715645881358028, 6.74294969418301968956156051808, 7.16908094728150537899855587269, 8.143980170472895790419522408103
