| L(s) = 1 | + 2-s − 2.61·3-s + 4-s − 0.740·5-s − 2.61·6-s − 2.94·7-s + 8-s + 3.81·9-s − 0.740·10-s + 3.23·11-s − 2.61·12-s + 2.51·13-s − 2.94·14-s + 1.93·15-s + 16-s − 1.46·17-s + 3.81·18-s − 1.61·19-s − 0.740·20-s + 7.69·21-s + 3.23·22-s + 4.93·23-s − 2.61·24-s − 4.45·25-s + 2.51·26-s − 2.13·27-s − 2.94·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.50·3-s + 0.5·4-s − 0.331·5-s − 1.06·6-s − 1.11·7-s + 0.353·8-s + 1.27·9-s − 0.234·10-s + 0.974·11-s − 0.753·12-s + 0.696·13-s − 0.787·14-s + 0.499·15-s + 0.250·16-s − 0.354·17-s + 0.900·18-s − 0.370·19-s − 0.165·20-s + 1.67·21-s + 0.688·22-s + 1.02·23-s − 0.533·24-s − 0.890·25-s + 0.492·26-s − 0.411·27-s − 0.556·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.217400646\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.217400646\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 2011 | \( 1 + T \) |
| good | 3 | \( 1 + 2.61T + 3T^{2} \) |
| 5 | \( 1 + 0.740T + 5T^{2} \) |
| 7 | \( 1 + 2.94T + 7T^{2} \) |
| 11 | \( 1 - 3.23T + 11T^{2} \) |
| 13 | \( 1 - 2.51T + 13T^{2} \) |
| 17 | \( 1 + 1.46T + 17T^{2} \) |
| 19 | \( 1 + 1.61T + 19T^{2} \) |
| 23 | \( 1 - 4.93T + 23T^{2} \) |
| 29 | \( 1 + 6.62T + 29T^{2} \) |
| 31 | \( 1 - 0.600T + 31T^{2} \) |
| 37 | \( 1 + 8.64T + 37T^{2} \) |
| 41 | \( 1 + 1.90T + 41T^{2} \) |
| 43 | \( 1 + 0.328T + 43T^{2} \) |
| 47 | \( 1 + 6.46T + 47T^{2} \) |
| 53 | \( 1 - 8.34T + 53T^{2} \) |
| 59 | \( 1 - 8.62T + 59T^{2} \) |
| 61 | \( 1 - 9.21T + 61T^{2} \) |
| 67 | \( 1 + 10.1T + 67T^{2} \) |
| 71 | \( 1 + 6.89T + 71T^{2} \) |
| 73 | \( 1 - 15.1T + 73T^{2} \) |
| 79 | \( 1 - 6.92T + 79T^{2} \) |
| 83 | \( 1 - 9.15T + 83T^{2} \) |
| 89 | \( 1 + 4.53T + 89T^{2} \) |
| 97 | \( 1 - 3.81T + 97T^{2} \) |
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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
−8.453563449244656366942168771703, −7.16856126924608387740576209950, −6.76818703102726204500666141612, −6.16466965751677460734332059790, −5.58720408436824183609336256094, −4.77847117412359373043584068219, −3.85854200487914043610130881243, −3.38535370661909139107073197244, −1.87366319499694542639523235334, −0.60830167669162817381016783715,
0.60830167669162817381016783715, 1.87366319499694542639523235334, 3.38535370661909139107073197244, 3.85854200487914043610130881243, 4.77847117412359373043584068219, 5.58720408436824183609336256094, 6.16466965751677460734332059790, 6.76818703102726204500666141612, 7.16856126924608387740576209950, 8.453563449244656366942168771703
