L(s) = 1 | + 2·2-s + 3·3-s + 4·4-s − 6.95·5-s + 6·6-s + 19.9·7-s + 8·8-s + 9·9-s − 13.9·10-s + 20.9·11-s + 12·12-s − 38.8·13-s + 39.9·14-s − 20.8·15-s + 16·16-s + 12·17-s + 18·18-s − 27.8·20-s + 59.8·21-s + 41.9·22-s + 29.0·23-s + 24·24-s − 76.6·25-s − 77.6·26-s + 27·27-s + 79.8·28-s + 10.0·29-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.622·5-s + 0.408·6-s + 1.07·7-s + 0.353·8-s + 0.333·9-s − 0.439·10-s + 0.574·11-s + 0.288·12-s − 0.828·13-s + 0.761·14-s − 0.359·15-s + 0.250·16-s + 0.171·17-s + 0.235·18-s − 0.311·20-s + 0.622·21-s + 0.406·22-s + 0.263·23-s + 0.204·24-s − 0.613·25-s − 0.585·26-s + 0.192·27-s + 0.538·28-s + 0.0646·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(4.977791540\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.977791540\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 2T \) |
| 3 | \( 1 - 3T \) |
| 19 | \( 1 \) |
good | 5 | \( 1 + 6.95T + 125T^{2} \) |
| 7 | \( 1 - 19.9T + 343T^{2} \) |
| 11 | \( 1 - 20.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 38.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 12T + 4.91e3T^{2} \) |
| 23 | \( 1 - 29.0T + 1.21e4T^{2} \) |
| 29 | \( 1 - 10.0T + 2.43e4T^{2} \) |
| 31 | \( 1 - 229.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 262.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 122.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 234.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 611.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 119.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 259.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 82.0T + 2.26e5T^{2} \) |
| 67 | \( 1 - 578.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 638.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 112.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 957.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 780.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 813.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.48e3T + 9.12e5T^{2} \) |
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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
−8.496072043955370132514445717173, −7.85701727124109511357969877063, −7.31315955764326640474729486729, −6.38852997389079820981637984850, −5.33701402625922782539267176050, −4.50270468945165135447002161485, −4.00092939256799134323568509442, −2.91020616747871398831128538625, −2.03873339809218803834537609057, −0.911773898876262887362997011410,
0.911773898876262887362997011410, 2.03873339809218803834537609057, 2.91020616747871398831128538625, 4.00092939256799134323568509442, 4.50270468945165135447002161485, 5.33701402625922782539267176050, 6.38852997389079820981637984850, 7.31315955764326640474729486729, 7.85701727124109511357969877063, 8.496072043955370132514445717173