| L(s) = 1 | − 1.28·3-s + 0.633·5-s − 3.19·7-s − 1.35·9-s + 2.65·11-s + 1.68·13-s − 0.811·15-s + 0.583·17-s + 6.55·19-s + 4.09·21-s − 6.42·23-s − 4.59·25-s + 5.58·27-s − 2.58·29-s − 4.14·31-s − 3.40·33-s − 2.02·35-s − 2.10·37-s − 2.15·39-s + 9.79·41-s + 8.03·43-s − 0.860·45-s + 3.69·47-s + 3.23·49-s − 0.747·51-s + 6.84·53-s + 1.68·55-s + ⋯ |
| L(s) = 1 | − 0.739·3-s + 0.283·5-s − 1.20·7-s − 0.452·9-s + 0.801·11-s + 0.467·13-s − 0.209·15-s + 0.141·17-s + 1.50·19-s + 0.894·21-s − 1.34·23-s − 0.919·25-s + 1.07·27-s − 0.480·29-s − 0.743·31-s − 0.592·33-s − 0.342·35-s − 0.346·37-s − 0.345·39-s + 1.52·41-s + 1.22·43-s − 0.128·45-s + 0.539·47-s + 0.462·49-s − 0.104·51-s + 0.939·53-s + 0.226·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 251 | \( 1 - T \) |
| good | 3 | \( 1 + 1.28T + 3T^{2} \) |
| 5 | \( 1 - 0.633T + 5T^{2} \) |
| 7 | \( 1 + 3.19T + 7T^{2} \) |
| 11 | \( 1 - 2.65T + 11T^{2} \) |
| 13 | \( 1 - 1.68T + 13T^{2} \) |
| 17 | \( 1 - 0.583T + 17T^{2} \) |
| 19 | \( 1 - 6.55T + 19T^{2} \) |
| 23 | \( 1 + 6.42T + 23T^{2} \) |
| 29 | \( 1 + 2.58T + 29T^{2} \) |
| 31 | \( 1 + 4.14T + 31T^{2} \) |
| 37 | \( 1 + 2.10T + 37T^{2} \) |
| 41 | \( 1 - 9.79T + 41T^{2} \) |
| 43 | \( 1 - 8.03T + 43T^{2} \) |
| 47 | \( 1 - 3.69T + 47T^{2} \) |
| 53 | \( 1 - 6.84T + 53T^{2} \) |
| 59 | \( 1 + 13.4T + 59T^{2} \) |
| 61 | \( 1 + 7.34T + 61T^{2} \) |
| 67 | \( 1 - 14.2T + 67T^{2} \) |
| 71 | \( 1 - 1.58T + 71T^{2} \) |
| 73 | \( 1 + 9.80T + 73T^{2} \) |
| 79 | \( 1 + 7.71T + 79T^{2} \) |
| 83 | \( 1 - 12.6T + 83T^{2} \) |
| 89 | \( 1 + 0.735T + 89T^{2} \) |
| 97 | \( 1 + 1.85T + 97T^{2} \) |
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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
−7.970343014638390022640855273236, −7.25329594370836364554052102855, −6.35922197509714487957060732189, −5.88758166604508453864968060217, −5.44687310918808112871937474567, −4.12982321476505406263580544927, −3.50884257582252310974205195605, −2.53826661927339437064877996362, −1.21776097376954244644323103062, 0,
1.21776097376954244644323103062, 2.53826661927339437064877996362, 3.50884257582252310974205195605, 4.12982321476505406263580544927, 5.44687310918808112871937474567, 5.88758166604508453864968060217, 6.35922197509714487957060732189, 7.25329594370836364554052102855, 7.970343014638390022640855273236
