| L(s) = 1 | + 1.41·2-s − 6.01·4-s + 4.37·5-s − 26.4·7-s − 19.7·8-s + 6.17·10-s + 11·11-s + 16.5·13-s − 37.3·14-s + 20.2·16-s + 58.2·17-s − 19·19-s − 26.3·20-s + 15.5·22-s + 72.7·23-s − 105.·25-s + 23.3·26-s + 159.·28-s + 219.·29-s + 136.·31-s + 186.·32-s + 82.0·34-s − 115.·35-s + 27.1·37-s − 26.7·38-s − 86.4·40-s − 186.·41-s + ⋯ |
| L(s) = 1 | + 0.498·2-s − 0.751·4-s + 0.391·5-s − 1.42·7-s − 0.873·8-s + 0.195·10-s + 0.301·11-s + 0.353·13-s − 0.712·14-s + 0.316·16-s + 0.830·17-s − 0.229·19-s − 0.294·20-s + 0.150·22-s + 0.659·23-s − 0.846·25-s + 0.176·26-s + 1.07·28-s + 1.40·29-s + 0.793·31-s + 1.03·32-s + 0.413·34-s − 0.559·35-s + 0.120·37-s − 0.114·38-s − 0.341·40-s − 0.711·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1881 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1881 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 - 11T \) |
| 19 | \( 1 + 19T \) |
| good | 2 | \( 1 - 1.41T + 8T^{2} \) |
| 5 | \( 1 - 4.37T + 125T^{2} \) |
| 7 | \( 1 + 26.4T + 343T^{2} \) |
| 13 | \( 1 - 16.5T + 2.19e3T^{2} \) |
| 17 | \( 1 - 58.2T + 4.91e3T^{2} \) |
| 23 | \( 1 - 72.7T + 1.21e4T^{2} \) |
| 29 | \( 1 - 219.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 136.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 27.1T + 5.06e4T^{2} \) |
| 41 | \( 1 + 186.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 415.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 414.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 589.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 331.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 725.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 731.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 242.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.08e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 321.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.02e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 516.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 703.T + 9.12e5T^{2} \) |
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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
−8.743545360021792555692198047903, −7.68475608422571834197560615375, −6.52663740148709035534364286266, −6.11837344777461770358008693540, −5.25382986486165065954185409266, −4.29103144834846050230803677715, −3.42554513204144043752753582577, −2.78781202182551367056612141747, −1.14056398284106399511530418745, 0,
1.14056398284106399511530418745, 2.78781202182551367056612141747, 3.42554513204144043752753582577, 4.29103144834846050230803677715, 5.25382986486165065954185409266, 6.11837344777461770358008693540, 6.52663740148709035534364286266, 7.68475608422571834197560615375, 8.743545360021792555692198047903
