| L(s) = 1 | − 1.41·2-s + 3·3-s − 5.99·4-s + 8.60·5-s − 4.25·6-s + 2.21·7-s + 19.8·8-s + 9·9-s − 12.2·10-s − 11·11-s − 17.9·12-s − 62.2·13-s − 3.14·14-s + 25.8·15-s + 19.8·16-s + 59.5·17-s − 12.7·18-s + 65.5·19-s − 51.5·20-s + 6.65·21-s + 15.5·22-s + 2.75·23-s + 59.4·24-s − 50.9·25-s + 88.1·26-s + 27·27-s − 13.2·28-s + ⋯ |
| L(s) = 1 | − 0.501·2-s + 0.577·3-s − 0.748·4-s + 0.769·5-s − 0.289·6-s + 0.119·7-s + 0.876·8-s + 0.333·9-s − 0.385·10-s − 0.301·11-s − 0.432·12-s − 1.32·13-s − 0.0600·14-s + 0.444·15-s + 0.309·16-s + 0.849·17-s − 0.167·18-s + 0.791·19-s − 0.576·20-s + 0.0691·21-s + 0.151·22-s + 0.0249·23-s + 0.506·24-s − 0.407·25-s + 0.665·26-s + 0.192·27-s − 0.0896·28-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 | \( 1 + 1.41T + 8T^{2} \) |
| 5 | \( 1 - 8.60T + 125T^{2} \) |
| 7 | \( 1 - 2.21T + 343T^{2} \) |
| 13 | \( 1 + 62.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 59.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 65.5T + 6.85e3T^{2} \) |
| 23 | \( 1 - 2.75T + 1.21e4T^{2} \) |
| 29 | \( 1 + 266.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 81.9T + 2.97e4T^{2} \) |
| 37 | \( 1 + 82.5T + 5.06e4T^{2} \) |
| 41 | \( 1 - 424.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 332.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 409.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 331.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 700.T + 2.05e5T^{2} \) |
| 67 | \( 1 + 520.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 582.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 526.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.17e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 161.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 982.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.40e3T + 9.12e5T^{2} \) |
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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
−8.489201283073800606508025827377, −7.60945835853108442667478866578, −7.31801830297086102930896978349, −5.82735063822508555770072148143, −5.21968083746149641794107228215, −4.34291801921353738007479766843, −3.28380648148925067414171344633, −2.23281991385902015140233194253, −1.28232170164513921595048891703, 0,
1.28232170164513921595048891703, 2.23281991385902015140233194253, 3.28380648148925067414171344633, 4.34291801921353738007479766843, 5.21968083746149641794107228215, 5.82735063822508555770072148143, 7.31801830297086102930896978349, 7.60945835853108442667478866578, 8.489201283073800606508025827377
