| L(s) = 1 | + 3.99·2-s − 3·3-s + 7.99·4-s + 5.91·5-s − 11.9·6-s + 20.5·7-s − 0.00499·8-s + 9·9-s + 23.6·10-s + 11·11-s − 23.9·12-s − 23.2·13-s + 82.2·14-s − 17.7·15-s − 64.0·16-s − 82.6·17-s + 35.9·18-s − 15.7·19-s + 47.3·20-s − 61.6·21-s + 43.9·22-s − 28.4·23-s + 0.0149·24-s − 89.9·25-s − 92.9·26-s − 27·27-s + 164.·28-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 0.577·3-s + 0.999·4-s + 0.529·5-s − 0.816·6-s + 1.10·7-s − 0.000220·8-s + 0.333·9-s + 0.748·10-s + 0.301·11-s − 0.577·12-s − 0.495·13-s + 1.56·14-s − 0.305·15-s − 1.00·16-s − 1.17·17-s + 0.471·18-s − 0.189·19-s + 0.529·20-s − 0.640·21-s + 0.426·22-s − 0.258·23-s + 0.000127·24-s − 0.719·25-s − 0.701·26-s − 0.192·27-s + 1.10·28-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
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 11 | \( 1 - 11T \) |
| 61 | \( 1 + 61T \) |
| good | 2 | \( 1 - 3.99T + 8T^{2} \) |
| 5 | \( 1 - 5.91T + 125T^{2} \) |
| 7 | \( 1 - 20.5T + 343T^{2} \) |
| 13 | \( 1 + 23.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 82.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 15.7T + 6.85e3T^{2} \) |
| 23 | \( 1 + 28.4T + 1.21e4T^{2} \) |
| 29 | \( 1 - 11.8T + 2.43e4T^{2} \) |
| 31 | \( 1 + 145.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 55.8T + 5.06e4T^{2} \) |
| 41 | \( 1 + 261.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 145.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 358.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 651.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 465.T + 2.05e5T^{2} \) |
| 67 | \( 1 - 541.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 621.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 448.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 371.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 218.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.32e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.50e3T + 9.12e5T^{2} \) |
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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
−8.367998448151198757473650436789, −7.26829864243008785181510390594, −6.54269428421077978118729027708, −5.77487472676858984019196884975, −5.09965379388447984262885830306, −4.50822268720543183225100379742, −3.73021361632144325087809362755, −2.38817543901759983931067704421, −1.67133279907210361489878520721, 0,
1.67133279907210361489878520721, 2.38817543901759983931067704421, 3.73021361632144325087809362755, 4.50822268720543183225100379742, 5.09965379388447984262885830306, 5.77487472676858984019196884975, 6.54269428421077978118729027708, 7.26829864243008785181510390594, 8.367998448151198757473650436789
