| L(s) = 1 | + 1.53·2-s + 3-s + 0.360·4-s + 1.53·6-s − 7-s − 2.51·8-s + 9-s + 4.33·11-s + 0.360·12-s − 13-s − 1.53·14-s − 4.59·16-s − 7.74·17-s + 1.53·18-s − 1.48·19-s − 21-s + 6.65·22-s + 4.03·23-s − 2.51·24-s − 1.53·26-s + 27-s − 0.360·28-s + 3.51·29-s − 5.72·31-s − 2.01·32-s + 4.33·33-s − 11.9·34-s + ⋯ |
| L(s) = 1 | + 1.08·2-s + 0.577·3-s + 0.180·4-s + 0.627·6-s − 0.377·7-s − 0.890·8-s + 0.333·9-s + 1.30·11-s + 0.104·12-s − 0.277·13-s − 0.410·14-s − 1.14·16-s − 1.87·17-s + 0.362·18-s − 0.339·19-s − 0.218·21-s + 1.41·22-s + 0.841·23-s − 0.514·24-s − 0.301·26-s + 0.192·27-s − 0.0682·28-s + 0.653·29-s − 1.02·31-s − 0.356·32-s + 0.754·33-s − 2.04·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6825 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 - 1.53T + 2T^{2} \) |
| 11 | \( 1 - 4.33T + 11T^{2} \) |
| 17 | \( 1 + 7.74T + 17T^{2} \) |
| 19 | \( 1 + 1.48T + 19T^{2} \) |
| 23 | \( 1 - 4.03T + 23T^{2} \) |
| 29 | \( 1 - 3.51T + 29T^{2} \) |
| 31 | \( 1 + 5.72T + 31T^{2} \) |
| 37 | \( 1 + 7.00T + 37T^{2} \) |
| 41 | \( 1 + 8.36T + 41T^{2} \) |
| 43 | \( 1 - 9.97T + 43T^{2} \) |
| 47 | \( 1 + 1.22T + 47T^{2} \) |
| 53 | \( 1 - 3.34T + 53T^{2} \) |
| 59 | \( 1 + 6.05T + 59T^{2} \) |
| 61 | \( 1 + 12.3T + 61T^{2} \) |
| 67 | \( 1 - 13.4T + 67T^{2} \) |
| 71 | \( 1 + 16.5T + 71T^{2} \) |
| 73 | \( 1 + 5.91T + 73T^{2} \) |
| 79 | \( 1 + 12.6T + 79T^{2} \) |
| 83 | \( 1 + 2.32T + 83T^{2} \) |
| 89 | \( 1 + 5.06T + 89T^{2} \) |
| 97 | \( 1 + 14.7T + 97T^{2} \) |
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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
−7.32302741996968932753607042255, −6.75177326167393116433936998126, −6.27666606034531285104358420495, −5.36077754618772949400046579000, −4.47953865599777302364629987911, −4.12472115013960831330111457859, −3.30375677971777727311093101910, −2.59763553997078142699621607556, −1.60137312869881220483968555195, 0,
1.60137312869881220483968555195, 2.59763553997078142699621607556, 3.30375677971777727311093101910, 4.12472115013960831330111457859, 4.47953865599777302364629987911, 5.36077754618772949400046579000, 6.27666606034531285104358420495, 6.75177326167393116433936998126, 7.32302741996968932753607042255
