| L(s) = 1 | − 2-s + 0.995·3-s + 4-s + 2.56·5-s − 0.995·6-s − 3.12·7-s − 8-s − 2.00·9-s − 2.56·10-s + 11-s + 0.995·12-s + 1.90·13-s + 3.12·14-s + 2.54·15-s + 16-s + 3.95·17-s + 2.00·18-s − 0.294·19-s + 2.56·20-s − 3.11·21-s − 22-s − 4.99·23-s − 0.995·24-s + 1.56·25-s − 1.90·26-s − 4.98·27-s − 3.12·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.574·3-s + 0.5·4-s + 1.14·5-s − 0.406·6-s − 1.18·7-s − 0.353·8-s − 0.669·9-s − 0.810·10-s + 0.301·11-s + 0.287·12-s + 0.528·13-s + 0.836·14-s + 0.658·15-s + 0.250·16-s + 0.959·17-s + 0.473·18-s − 0.0675·19-s + 0.573·20-s − 0.679·21-s − 0.213·22-s − 1.04·23-s − 0.203·24-s + 0.313·25-s − 0.373·26-s − 0.959·27-s − 0.591·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 197 | \( 1 - T \) |
| good | 3 | \( 1 - 0.995T + 3T^{2} \) |
| 5 | \( 1 - 2.56T + 5T^{2} \) |
| 7 | \( 1 + 3.12T + 7T^{2} \) |
| 13 | \( 1 - 1.90T + 13T^{2} \) |
| 17 | \( 1 - 3.95T + 17T^{2} \) |
| 19 | \( 1 + 0.294T + 19T^{2} \) |
| 23 | \( 1 + 4.99T + 23T^{2} \) |
| 29 | \( 1 + 6.71T + 29T^{2} \) |
| 31 | \( 1 + 2.58T + 31T^{2} \) |
| 37 | \( 1 + 8.71T + 37T^{2} \) |
| 41 | \( 1 + 2.08T + 41T^{2} \) |
| 43 | \( 1 - 3.12T + 43T^{2} \) |
| 47 | \( 1 - 5.20T + 47T^{2} \) |
| 53 | \( 1 + 3.14T + 53T^{2} \) |
| 59 | \( 1 + 6.61T + 59T^{2} \) |
| 61 | \( 1 - 6.08T + 61T^{2} \) |
| 67 | \( 1 - 1.50T + 67T^{2} \) |
| 71 | \( 1 + 3.24T + 71T^{2} \) |
| 73 | \( 1 + 10.5T + 73T^{2} \) |
| 79 | \( 1 - 5.87T + 79T^{2} \) |
| 83 | \( 1 - 11.7T + 83T^{2} \) |
| 89 | \( 1 + 8.04T + 89T^{2} \) |
| 97 | \( 1 + 16.7T + 97T^{2} \) |
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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
−8.154609365512655317371051353920, −7.37916012760467014404221994426, −6.52211587471098224025950824985, −5.87141546224217172896902864415, −5.48953056727595857270741468852, −3.80895073731064654073166756420, −3.24652041363380930286754716234, −2.32729972134856006642605988079, −1.53180774462012345684831564331, 0,
1.53180774462012345684831564331, 2.32729972134856006642605988079, 3.24652041363380930286754716234, 3.80895073731064654073166756420, 5.48953056727595857270741468852, 5.87141546224217172896902864415, 6.52211587471098224025950824985, 7.37916012760467014404221994426, 8.154609365512655317371051353920
