| L(s) = 1 | − 2-s − 0.154·3-s + 4-s − 5-s + 0.154·6-s − 4.41·7-s − 8-s − 2.97·9-s + 10-s − 3.07·11-s − 0.154·12-s + 6.08·13-s + 4.41·14-s + 0.154·15-s + 16-s + 6.99·17-s + 2.97·18-s − 0.745·19-s − 20-s + 0.683·21-s + 3.07·22-s − 2.22·23-s + 0.154·24-s + 25-s − 6.08·26-s + 0.925·27-s − 4.41·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.0893·3-s + 0.5·4-s − 0.447·5-s + 0.0631·6-s − 1.66·7-s − 0.353·8-s − 0.992·9-s + 0.316·10-s − 0.926·11-s − 0.0446·12-s + 1.68·13-s + 1.18·14-s + 0.0399·15-s + 0.250·16-s + 1.69·17-s + 0.701·18-s − 0.171·19-s − 0.223·20-s + 0.149·21-s + 0.654·22-s − 0.463·23-s + 0.0315·24-s + 0.200·25-s − 1.19·26-s + 0.178·27-s − 0.834·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 + T \) |
| good | 3 | \( 1 + 0.154T + 3T^{2} \) |
| 7 | \( 1 + 4.41T + 7T^{2} \) |
| 11 | \( 1 + 3.07T + 11T^{2} \) |
| 13 | \( 1 - 6.08T + 13T^{2} \) |
| 17 | \( 1 - 6.99T + 17T^{2} \) |
| 19 | \( 1 + 0.745T + 19T^{2} \) |
| 23 | \( 1 + 2.22T + 23T^{2} \) |
| 29 | \( 1 - 2.48T + 29T^{2} \) |
| 31 | \( 1 - 2.52T + 31T^{2} \) |
| 37 | \( 1 + 3.01T + 37T^{2} \) |
| 41 | \( 1 - 6.07T + 41T^{2} \) |
| 43 | \( 1 - 1.67T + 43T^{2} \) |
| 47 | \( 1 - 2.53T + 47T^{2} \) |
| 53 | \( 1 + 0.745T + 53T^{2} \) |
| 59 | \( 1 + 3.79T + 59T^{2} \) |
| 61 | \( 1 - 4.16T + 61T^{2} \) |
| 67 | \( 1 + 13.0T + 67T^{2} \) |
| 71 | \( 1 - 8.13T + 71T^{2} \) |
| 73 | \( 1 - 1.02T + 73T^{2} \) |
| 79 | \( 1 + 7.95T + 79T^{2} \) |
| 83 | \( 1 + 6.71T + 83T^{2} \) |
| 89 | \( 1 - 8.62T + 89T^{2} \) |
| 97 | \( 1 + 16.4T + 97T^{2} \) |
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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
−8.166745841704654286958969829535, −7.52726226094679819796355127152, −6.54778891767769567222368734758, −5.98037649564794678744286078589, −5.46417807191464345466154940162, −3.93932781410379138043220494927, −3.21750902013406720900140277475, −2.70165191426004538786269852196, −1.04964725766603752685732044786, 0,
1.04964725766603752685732044786, 2.70165191426004538786269852196, 3.21750902013406720900140277475, 3.93932781410379138043220494927, 5.46417807191464345466154940162, 5.98037649564794678744286078589, 6.54778891767769567222368734758, 7.52726226094679819796355127152, 8.166745841704654286958969829535
