| L(s) = 1 | − 2-s − 1.11·3-s + 4-s − 2.63·5-s + 1.11·6-s − 4.04·7-s − 8-s − 1.76·9-s + 2.63·10-s − 0.240·11-s − 1.11·12-s − 4.90·13-s + 4.04·14-s + 2.93·15-s + 16-s − 2.68·17-s + 1.76·18-s − 19-s − 2.63·20-s + 4.49·21-s + 0.240·22-s + 4.56·23-s + 1.11·24-s + 1.95·25-s + 4.90·26-s + 5.29·27-s − 4.04·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.642·3-s + 0.5·4-s − 1.17·5-s + 0.454·6-s − 1.52·7-s − 0.353·8-s − 0.587·9-s + 0.834·10-s − 0.0723·11-s − 0.321·12-s − 1.35·13-s + 1.08·14-s + 0.757·15-s + 0.250·16-s − 0.650·17-s + 0.415·18-s − 0.229·19-s − 0.589·20-s + 0.981·21-s + 0.0511·22-s + 0.950·23-s + 0.227·24-s + 0.391·25-s + 0.961·26-s + 1.01·27-s − 0.763·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 211 | \( 1 + T \) |
| good | 3 | \( 1 + 1.11T + 3T^{2} \) |
| 5 | \( 1 + 2.63T + 5T^{2} \) |
| 7 | \( 1 + 4.04T + 7T^{2} \) |
| 11 | \( 1 + 0.240T + 11T^{2} \) |
| 13 | \( 1 + 4.90T + 13T^{2} \) |
| 17 | \( 1 + 2.68T + 17T^{2} \) |
| 23 | \( 1 - 4.56T + 23T^{2} \) |
| 29 | \( 1 + 3.19T + 29T^{2} \) |
| 31 | \( 1 - 4.37T + 31T^{2} \) |
| 37 | \( 1 + 9.88T + 37T^{2} \) |
| 41 | \( 1 + 0.240T + 41T^{2} \) |
| 43 | \( 1 - 11.1T + 43T^{2} \) |
| 47 | \( 1 - 5.99T + 47T^{2} \) |
| 53 | \( 1 + 11.9T + 53T^{2} \) |
| 59 | \( 1 + 3.67T + 59T^{2} \) |
| 61 | \( 1 - 4.10T + 61T^{2} \) |
| 67 | \( 1 - 13.1T + 67T^{2} \) |
| 71 | \( 1 - 6.19T + 71T^{2} \) |
| 73 | \( 1 - 0.320T + 73T^{2} \) |
| 79 | \( 1 - 3.48T + 79T^{2} \) |
| 83 | \( 1 - 1.60T + 83T^{2} \) |
| 89 | \( 1 - 1.30T + 89T^{2} \) |
| 97 | \( 1 + 15.2T + 97T^{2} \) |
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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
−7.40444357107755728199389842839, −6.87067106761113287815653888705, −6.35637040665767656216769496043, −5.48862194786259432532128687461, −4.72385342721215917125783495522, −3.77173064684126166156672573543, −3.03000154402076250805665005890, −2.34881105300437848894740270927, −0.62324847783582474252169983077, 0,
0.62324847783582474252169983077, 2.34881105300437848894740270927, 3.03000154402076250805665005890, 3.77173064684126166156672573543, 4.72385342721215917125783495522, 5.48862194786259432532128687461, 6.35637040665767656216769496043, 6.87067106761113287815653888705, 7.40444357107755728199389842839
