| L(s) = 1 | − 3-s − 2·4-s − 5-s − 3·7-s − 2·9-s + 11-s + 2·12-s + 13-s + 15-s + 4·16-s − 3·17-s + 6·19-s + 2·20-s + 3·21-s − 7·23-s + 25-s + 5·27-s + 6·28-s + 6·29-s + 2·31-s − 33-s + 3·35-s + 4·36-s + 2·37-s − 39-s + 6·41-s + 11·43-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s − 0.447·5-s − 1.13·7-s − 2/3·9-s + 0.301·11-s + 0.577·12-s + 0.277·13-s + 0.258·15-s + 16-s − 0.727·17-s + 1.37·19-s + 0.447·20-s + 0.654·21-s − 1.45·23-s + 1/5·25-s + 0.962·27-s + 1.13·28-s + 1.11·29-s + 0.359·31-s − 0.174·33-s + 0.507·35-s + 2/3·36-s + 0.328·37-s − 0.160·39-s + 0.937·41-s + 1.67·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 73 | \( 1 - T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 3 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−8.153589475422444686265172960536, −7.41391883485190109017556912316, −6.26535782675157793867525284266, −6.04734528172694599530432391524, −5.04269085383946209898777004120, −4.28833913711699395703666674612, −3.51085337704015508686114321282, −2.73246890332051102726427407766, −0.942555327800869250759451453100, 0,
0.942555327800869250759451453100, 2.73246890332051102726427407766, 3.51085337704015508686114321282, 4.28833913711699395703666674612, 5.04269085383946209898777004120, 6.04734528172694599530432391524, 6.26535782675157793867525284266, 7.41391883485190109017556912316, 8.153589475422444686265172960536
