| L(s) = 1 | − 5.12·2-s + 3·3-s + 18.2·4-s − 6.55·5-s − 15.3·6-s + 17.3·7-s − 52.7·8-s + 9·9-s + 33.6·10-s − 19.9·11-s + 54.8·12-s + 63.2·13-s − 89.2·14-s − 19.6·15-s + 124.·16-s − 116.·17-s − 46.1·18-s + 104.·19-s − 119.·20-s + 52.1·21-s + 102.·22-s + 23·23-s − 158.·24-s − 82.0·25-s − 324.·26-s + 27·27-s + 318.·28-s + ⋯ |
| L(s) = 1 | − 1.81·2-s + 0.577·3-s + 2.28·4-s − 0.586·5-s − 1.04·6-s + 0.939·7-s − 2.33·8-s + 0.333·9-s + 1.06·10-s − 0.545·11-s + 1.32·12-s + 1.34·13-s − 1.70·14-s − 0.338·15-s + 1.94·16-s − 1.66·17-s − 0.604·18-s + 1.26·19-s − 1.34·20-s + 0.542·21-s + 0.989·22-s + 0.208·23-s − 1.34·24-s − 0.656·25-s − 2.44·26-s + 0.192·27-s + 2.14·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.109731700\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.109731700\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 23 | \( 1 - 23T \) |
| 29 | \( 1 + 29T \) |
| good | 2 | \( 1 + 5.12T + 8T^{2} \) |
| 5 | \( 1 + 6.55T + 125T^{2} \) |
| 7 | \( 1 - 17.3T + 343T^{2} \) |
| 11 | \( 1 + 19.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 63.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 116.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 104.T + 6.85e3T^{2} \) |
| 31 | \( 1 - 166.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 78.3T + 5.06e4T^{2} \) |
| 41 | \( 1 + 81.6T + 6.89e4T^{2} \) |
| 43 | \( 1 + 56.9T + 7.95e4T^{2} \) |
| 47 | \( 1 - 252.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 349.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 626.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 38.2T + 2.26e5T^{2} \) |
| 67 | \( 1 + 657.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 817.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 215.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.07e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 131.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.34e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 295.T + 9.12e5T^{2} \) |
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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
−8.679774200296797640545857615404, −8.231291653011885985869248318593, −7.58962760549353835248255787346, −6.94959369594388091313739991360, −5.95785534437225296565606175117, −4.67680094189141555712502068264, −3.55660197573070433187853629013, −2.45436364349967187344506126005, −1.59338597023448186737560882287, −0.64339483129564521209288777769,
0.64339483129564521209288777769, 1.59338597023448186737560882287, 2.45436364349967187344506126005, 3.55660197573070433187853629013, 4.67680094189141555712502068264, 5.95785534437225296565606175117, 6.94959369594388091313739991360, 7.58962760549353835248255787346, 8.231291653011885985869248318593, 8.679774200296797640545857615404
