| L(s) = 1 | + 2-s − 1.41·3-s + 4-s + 2.76·5-s − 1.41·6-s − 4.92·7-s + 8-s − 0.984·9-s + 2.76·10-s + 2.50·11-s − 1.41·12-s − 1.97·13-s − 4.92·14-s − 3.92·15-s + 16-s − 4.79·17-s − 0.984·18-s − 3.03·19-s + 2.76·20-s + 6.99·21-s + 2.50·22-s + 1.77·23-s − 1.41·24-s + 2.65·25-s − 1.97·26-s + 5.65·27-s − 4.92·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.819·3-s + 0.5·4-s + 1.23·5-s − 0.579·6-s − 1.86·7-s + 0.353·8-s − 0.328·9-s + 0.874·10-s + 0.754·11-s − 0.409·12-s − 0.548·13-s − 1.31·14-s − 1.01·15-s + 0.250·16-s − 1.16·17-s − 0.231·18-s − 0.695·19-s + 0.618·20-s + 1.52·21-s + 0.533·22-s + 0.369·23-s − 0.289·24-s + 0.530·25-s − 0.387·26-s + 1.08·27-s − 0.930·28-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
| \(L(1)\) |
\(\approx\) |
\(1.929912032\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.929912032\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 1.41T + 3T^{2} \) |
| 5 | \( 1 - 2.76T + 5T^{2} \) |
| 7 | \( 1 + 4.92T + 7T^{2} \) |
| 11 | \( 1 - 2.50T + 11T^{2} \) |
| 13 | \( 1 + 1.97T + 13T^{2} \) |
| 17 | \( 1 + 4.79T + 17T^{2} \) |
| 19 | \( 1 + 3.03T + 19T^{2} \) |
| 23 | \( 1 - 1.77T + 23T^{2} \) |
| 29 | \( 1 - 0.104T + 29T^{2} \) |
| 31 | \( 1 - 7.96T + 31T^{2} \) |
| 37 | \( 1 - 7.75T + 37T^{2} \) |
| 41 | \( 1 + 1.20T + 41T^{2} \) |
| 43 | \( 1 - 5.72T + 43T^{2} \) |
| 47 | \( 1 - 10.5T + 47T^{2} \) |
| 53 | \( 1 - 0.555T + 53T^{2} \) |
| 59 | \( 1 - 12.3T + 59T^{2} \) |
| 61 | \( 1 + 10.7T + 61T^{2} \) |
| 67 | \( 1 - 4.59T + 67T^{2} \) |
| 71 | \( 1 + 2.55T + 71T^{2} \) |
| 73 | \( 1 - 5.62T + 73T^{2} \) |
| 79 | \( 1 + 11.4T + 79T^{2} \) |
| 83 | \( 1 - 10.5T + 83T^{2} \) |
| 89 | \( 1 + 3.63T + 89T^{2} \) |
| 97 | \( 1 - 0.573T + 97T^{2} \) |
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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
−8.711807368640517380283046560624, −7.23009365058157528897441769161, −6.50493814169975162022373045401, −6.21642101143380178414800234244, −5.75755448679539682658859048913, −4.76111447665721307369406939716, −3.96652926934314445509111636915, −2.80863715332010582494755390563, −2.32465912197749071729181447496, −0.71432607705714294208870183027,
0.71432607705714294208870183027, 2.32465912197749071729181447496, 2.80863715332010582494755390563, 3.96652926934314445509111636915, 4.76111447665721307369406939716, 5.75755448679539682658859048913, 6.21642101143380178414800234244, 6.50493814169975162022373045401, 7.23009365058157528897441769161, 8.711807368640517380283046560624
