| L(s) = 1 | + 5.32·2-s + 3·3-s + 20.4·4-s − 19.7·5-s + 15.9·6-s + 5.38·7-s + 66.0·8-s + 9·9-s − 105.·10-s − 11·11-s + 61.2·12-s − 16.9·13-s + 28.6·14-s − 59.3·15-s + 188.·16-s + 20.3·17-s + 47.9·18-s + 90.6·19-s − 403.·20-s + 16.1·21-s − 58.6·22-s − 41.7·23-s + 198.·24-s + 266.·25-s − 90.3·26-s + 27·27-s + 109.·28-s + ⋯ |
| L(s) = 1 | + 1.88·2-s + 0.577·3-s + 2.55·4-s − 1.76·5-s + 1.08·6-s + 0.290·7-s + 2.92·8-s + 0.333·9-s − 3.33·10-s − 0.301·11-s + 1.47·12-s − 0.361·13-s + 0.547·14-s − 1.02·15-s + 2.95·16-s + 0.289·17-s + 0.628·18-s + 1.09·19-s − 4.51·20-s + 0.167·21-s − 0.568·22-s − 0.378·23-s + 1.68·24-s + 2.13·25-s − 0.681·26-s + 0.192·27-s + 0.741·28-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
| \(L(2)\) |
\(\approx\) |
\(7.998907231\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.998907231\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 5.32T + 8T^{2} \) |
| 5 | \( 1 + 19.7T + 125T^{2} \) |
| 7 | \( 1 - 5.38T + 343T^{2} \) |
| 13 | \( 1 + 16.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 20.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 90.6T + 6.85e3T^{2} \) |
| 23 | \( 1 + 41.7T + 1.21e4T^{2} \) |
| 29 | \( 1 - 103.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 122.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 424.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 435.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 211.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 72.6T + 1.03e5T^{2} \) |
| 53 | \( 1 - 172.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 495.T + 2.05e5T^{2} \) |
| 67 | \( 1 + 237.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 571.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 444.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 544.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 969.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 621.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.38e3T + 9.12e5T^{2} \) |
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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
−8.306249879957745254116155314653, −7.73063189678149139387346886047, −7.28175655902063348637949828881, −6.34249939726097292204110838749, −5.25046210106603308379772594758, −4.51051818000018565793418357881, −3.97553561799039066880391241341, −3.14706639819649368359915128103, −2.52442261896631537757064881395, −0.976480761639984143044002078863,
0.976480761639984143044002078863, 2.52442261896631537757064881395, 3.14706639819649368359915128103, 3.97553561799039066880391241341, 4.51051818000018565793418357881, 5.25046210106603308379772594758, 6.34249939726097292204110838749, 7.28175655902063348637949828881, 7.73063189678149139387346886047, 8.306249879957745254116155314653
