| L(s) = 1 | + 0.736·2-s − 3-s − 1.45·4-s − 2.90·5-s − 0.736·6-s − 3.09·7-s − 2.54·8-s + 9-s − 2.14·10-s + 4.62·11-s + 1.45·12-s − 1.53·13-s − 2.27·14-s + 2.90·15-s + 1.03·16-s + 17-s + 0.736·18-s − 1.13·19-s + 4.23·20-s + 3.09·21-s + 3.40·22-s + 2.40·23-s + 2.54·24-s + 3.45·25-s − 1.13·26-s − 27-s + 4.50·28-s + ⋯ |
| L(s) = 1 | + 0.520·2-s − 0.577·3-s − 0.728·4-s − 1.30·5-s − 0.300·6-s − 1.16·7-s − 0.900·8-s + 0.333·9-s − 0.677·10-s + 1.39·11-s + 0.420·12-s − 0.426·13-s − 0.608·14-s + 0.750·15-s + 0.259·16-s + 0.242·17-s + 0.173·18-s − 0.259·19-s + 0.947·20-s + 0.674·21-s + 0.725·22-s + 0.502·23-s + 0.519·24-s + 0.691·25-s − 0.222·26-s − 0.192·27-s + 0.851·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 79 | \( 1 - T \) |
| good | 2 | \( 1 - 0.736T + 2T^{2} \) |
| 5 | \( 1 + 2.90T + 5T^{2} \) |
| 7 | \( 1 + 3.09T + 7T^{2} \) |
| 11 | \( 1 - 4.62T + 11T^{2} \) |
| 13 | \( 1 + 1.53T + 13T^{2} \) |
| 19 | \( 1 + 1.13T + 19T^{2} \) |
| 23 | \( 1 - 2.40T + 23T^{2} \) |
| 29 | \( 1 - 6.76T + 29T^{2} \) |
| 31 | \( 1 - 4.89T + 31T^{2} \) |
| 37 | \( 1 - 9.35T + 37T^{2} \) |
| 41 | \( 1 + 9.50T + 41T^{2} \) |
| 43 | \( 1 - 7.79T + 43T^{2} \) |
| 47 | \( 1 + 6.64T + 47T^{2} \) |
| 53 | \( 1 + 5.07T + 53T^{2} \) |
| 59 | \( 1 + 5.84T + 59T^{2} \) |
| 61 | \( 1 + 12.9T + 61T^{2} \) |
| 67 | \( 1 - 11.6T + 67T^{2} \) |
| 71 | \( 1 + 12.1T + 71T^{2} \) |
| 73 | \( 1 + 4.49T + 73T^{2} \) |
| 83 | \( 1 - 1.03T + 83T^{2} \) |
| 89 | \( 1 - 0.673T + 89T^{2} \) |
| 97 | \( 1 + 16.0T + 97T^{2} \) |
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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
−8.097836708112421643489463590998, −7.18795826415079487829340370858, −6.42284813798675217491893491640, −6.01052165035877127437011435787, −4.69796619991179060877134026913, −4.42214019095324602662106316274, −3.53561895280472824821061317909, −2.99116000589280667731864790387, −0.997877406304356421471046860515, 0,
0.997877406304356421471046860515, 2.99116000589280667731864790387, 3.53561895280472824821061317909, 4.42214019095324602662106316274, 4.69796619991179060877134026913, 6.01052165035877127437011435787, 6.42284813798675217491893491640, 7.18795826415079487829340370858, 8.097836708112421643489463590998
