| L(s) = 1 | − 2·2-s + 3.24·3-s + 4·4-s − 5.66·5-s − 6.49·6-s − 8·8-s − 16.4·9-s + 11.3·10-s − 25.6·11-s + 12.9·12-s + 16.1·13-s − 18.3·15-s + 16·16-s + 25.7·17-s + 32.8·18-s − 56.2·19-s − 22.6·20-s + 51.3·22-s + 23·23-s − 25.9·24-s − 92.9·25-s − 32.3·26-s − 141.·27-s + 219.·29-s + 36.7·30-s + 100.·31-s − 32·32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.625·3-s + 0.5·4-s − 0.506·5-s − 0.442·6-s − 0.353·8-s − 0.609·9-s + 0.358·10-s − 0.703·11-s + 0.312·12-s + 0.345·13-s − 0.316·15-s + 0.250·16-s + 0.367·17-s + 0.430·18-s − 0.679·19-s − 0.253·20-s + 0.497·22-s + 0.208·23-s − 0.221·24-s − 0.743·25-s − 0.244·26-s − 1.00·27-s + 1.40·29-s + 0.223·30-s + 0.583·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2254 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2254 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.065791303\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.065791303\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 7 | \( 1 \) |
| 23 | \( 1 - 23T \) |
| good | 3 | \( 1 - 3.24T + 27T^{2} \) |
| 5 | \( 1 + 5.66T + 125T^{2} \) |
| 11 | \( 1 + 25.6T + 1.33e3T^{2} \) |
| 13 | \( 1 - 16.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 25.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 56.2T + 6.85e3T^{2} \) |
| 29 | \( 1 - 219.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 100.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 151.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 277.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 146.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 509.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 360.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 66.5T + 2.05e5T^{2} \) |
| 61 | \( 1 + 71.1T + 2.26e5T^{2} \) |
| 67 | \( 1 + 77.6T + 3.00e5T^{2} \) |
| 71 | \( 1 + 592.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 125.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 658.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.12e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 99.7T + 7.04e5T^{2} \) |
| 97 | \( 1 + 261.T + 9.12e5T^{2} \) |
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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
−8.370848131366736543077371212968, −8.274008513827753900045628768247, −7.39878575262768386806998596925, −6.49823193081186354564047471033, −5.65534487887244704424521382694, −4.61212292198377991660406948209, −3.48173125629391957163520841663, −2.80903280393059756042739053016, −1.82220469328173264195004817339, −0.48280472632373449643976646562,
0.48280472632373449643976646562, 1.82220469328173264195004817339, 2.80903280393059756042739053016, 3.48173125629391957163520841663, 4.61212292198377991660406948209, 5.65534487887244704424521382694, 6.49823193081186354564047471033, 7.39878575262768386806998596925, 8.274008513827753900045628768247, 8.370848131366736543077371212968
