| L(s) = 1 | − 2-s + 4-s + 3.24·5-s − 3.10·7-s − 8-s − 3.24·10-s + 11-s − 1.57·13-s + 3.10·14-s + 16-s − 7.38·17-s + 0.856·19-s + 3.24·20-s − 22-s − 3.32·23-s + 5.52·25-s + 1.57·26-s − 3.10·28-s − 8.91·29-s + 6.52·31-s − 32-s + 7.38·34-s − 10.0·35-s − 8.77·37-s − 0.856·38-s − 3.24·40-s − 5.10·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.45·5-s − 1.17·7-s − 0.353·8-s − 1.02·10-s + 0.301·11-s − 0.435·13-s + 0.828·14-s + 0.250·16-s − 1.79·17-s + 0.196·19-s + 0.725·20-s − 0.213·22-s − 0.693·23-s + 1.10·25-s + 0.308·26-s − 0.586·28-s − 1.65·29-s + 1.17·31-s − 0.176·32-s + 1.26·34-s − 1.70·35-s − 1.44·37-s − 0.138·38-s − 0.513·40-s − 0.796·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1782 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1782 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 5 | \( 1 - 3.24T + 5T^{2} \) |
| 7 | \( 1 + 3.10T + 7T^{2} \) |
| 13 | \( 1 + 1.57T + 13T^{2} \) |
| 17 | \( 1 + 7.38T + 17T^{2} \) |
| 19 | \( 1 - 0.856T + 19T^{2} \) |
| 23 | \( 1 + 3.32T + 23T^{2} \) |
| 29 | \( 1 + 8.91T + 29T^{2} \) |
| 31 | \( 1 - 6.52T + 31T^{2} \) |
| 37 | \( 1 + 8.77T + 37T^{2} \) |
| 41 | \( 1 + 5.10T + 41T^{2} \) |
| 43 | \( 1 - 3.77T + 43T^{2} \) |
| 47 | \( 1 + 13.2T + 47T^{2} \) |
| 53 | \( 1 - 7.06T + 53T^{2} \) |
| 59 | \( 1 - 4.28T + 59T^{2} \) |
| 61 | \( 1 + 9.34T + 61T^{2} \) |
| 67 | \( 1 - 0.773T + 67T^{2} \) |
| 71 | \( 1 - 0.816T + 71T^{2} \) |
| 73 | \( 1 - 9.87T + 73T^{2} \) |
| 79 | \( 1 - 4.53T + 79T^{2} \) |
| 83 | \( 1 + 1.14T + 83T^{2} \) |
| 89 | \( 1 - 2.67T + 89T^{2} \) |
| 97 | \( 1 + 2.14T + 97T^{2} \) |
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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
−9.166326293183100860467075399984, −8.351214509961746935569285182534, −7.09324302027837994827250931221, −6.53345817515477166530886480714, −5.97353750444219536119305965378, −4.96482293608452616939470503521, −3.62150185290680381200526483709, −2.48766289713268609973104138265, −1.76210695117849676519215505410, 0,
1.76210695117849676519215505410, 2.48766289713268609973104138265, 3.62150185290680381200526483709, 4.96482293608452616939470503521, 5.97353750444219536119305965378, 6.53345817515477166530886480714, 7.09324302027837994827250931221, 8.351214509961746935569285182534, 9.166326293183100860467075399984
