| L(s) = 1 | − 1.73·2-s − 1.78·3-s + 1.00·4-s − 2.76·5-s + 3.09·6-s − 4.11·7-s + 1.72·8-s + 0.197·9-s + 4.79·10-s − 1.99·11-s − 1.79·12-s − 4.67·13-s + 7.13·14-s + 4.94·15-s − 4.99·16-s − 2.57·17-s − 0.341·18-s − 2.14·19-s − 2.77·20-s + 7.36·21-s + 3.45·22-s − 1.75·23-s − 3.08·24-s + 2.64·25-s + 8.10·26-s + 5.01·27-s − 4.13·28-s + ⋯ |
| L(s) = 1 | − 1.22·2-s − 1.03·3-s + 0.502·4-s − 1.23·5-s + 1.26·6-s − 1.55·7-s + 0.609·8-s + 0.0657·9-s + 1.51·10-s − 0.600·11-s − 0.518·12-s − 1.29·13-s + 1.90·14-s + 1.27·15-s − 1.24·16-s − 0.625·17-s − 0.0805·18-s − 0.492·19-s − 0.621·20-s + 1.60·21-s + 0.735·22-s − 0.365·23-s − 0.629·24-s + 0.529·25-s + 1.58·26-s + 0.964·27-s − 0.781·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4031 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4031 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 29 | \( 1 + T \) |
| 139 | \( 1 + T \) |
| good | 2 | \( 1 + 1.73T + 2T^{2} \) |
| 3 | \( 1 + 1.78T + 3T^{2} \) |
| 5 | \( 1 + 2.76T + 5T^{2} \) |
| 7 | \( 1 + 4.11T + 7T^{2} \) |
| 11 | \( 1 + 1.99T + 11T^{2} \) |
| 13 | \( 1 + 4.67T + 13T^{2} \) |
| 17 | \( 1 + 2.57T + 17T^{2} \) |
| 19 | \( 1 + 2.14T + 19T^{2} \) |
| 23 | \( 1 + 1.75T + 23T^{2} \) |
| 31 | \( 1 - 6.89T + 31T^{2} \) |
| 37 | \( 1 - 5.25T + 37T^{2} \) |
| 41 | \( 1 + 6.91T + 41T^{2} \) |
| 43 | \( 1 + 4.92T + 43T^{2} \) |
| 47 | \( 1 + 4.03T + 47T^{2} \) |
| 53 | \( 1 - 6.90T + 53T^{2} \) |
| 59 | \( 1 + 11.1T + 59T^{2} \) |
| 61 | \( 1 + 3.54T + 61T^{2} \) |
| 67 | \( 1 - 8.82T + 67T^{2} \) |
| 71 | \( 1 - 1.32T + 71T^{2} \) |
| 73 | \( 1 - 1.21T + 73T^{2} \) |
| 79 | \( 1 + 13.8T + 79T^{2} \) |
| 83 | \( 1 - 14.0T + 83T^{2} \) |
| 89 | \( 1 - 7.91T + 89T^{2} \) |
| 97 | \( 1 - 17.0T + 97T^{2} \) |
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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
−8.074253517332562636693744430555, −7.44499362178655407156452596764, −6.72660844509197914369644887513, −6.19374111338002442192057961209, −4.99053782184400409331493233227, −4.42564394226643200903309280659, −3.32151017944652643597340662433, −2.34630957423467130967693507729, −0.53833120869434795336297934640, 0,
0.53833120869434795336297934640, 2.34630957423467130967693507729, 3.32151017944652643597340662433, 4.42564394226643200903309280659, 4.99053782184400409331493233227, 6.19374111338002442192057961209, 6.72660844509197914369644887513, 7.44499362178655407156452596764, 8.074253517332562636693744430555
