L(s) = 1 | + 0.517i·2-s − i·3-s + 1.73·4-s + i·5-s + 0.517·6-s + (2.63 − 0.189i)7-s + 1.93i·8-s − 9-s − 0.517·10-s + (−3 + 1.41i)11-s − 1.73i·12-s + 4.24·13-s + (0.0980 + 1.36i)14-s + 15-s + 2.46·16-s + 2.44·17-s + ⋯ |
L(s) = 1 | + 0.366i·2-s − 0.577i·3-s + 0.866·4-s + 0.447i·5-s + 0.211·6-s + (0.997 − 0.0716i)7-s + 0.683i·8-s − 0.333·9-s − 0.163·10-s + (−0.904 + 0.426i)11-s − 0.499i·12-s + 1.17·13-s + (0.0262 + 0.365i)14-s + 0.258·15-s + 0.616·16-s + 0.594·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.871 - 0.490i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.871 - 0.490i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.293097059\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.293097059\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + iT \) |
| 5 | \( 1 - iT \) |
| 7 | \( 1 + (-2.63 + 0.189i)T \) |
| 11 | \( 1 + (3 - 1.41i)T \) |
good | 2 | \( 1 - 0.517iT - 2T^{2} \) |
| 13 | \( 1 - 4.24T + 13T^{2} \) |
| 17 | \( 1 - 2.44T + 17T^{2} \) |
| 19 | \( 1 - 1.03T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 - 1.03iT - 29T^{2} \) |
| 31 | \( 1 - 2.92iT - 31T^{2} \) |
| 37 | \( 1 + 0.535T + 37T^{2} \) |
| 41 | \( 1 - 6.69T + 41T^{2} \) |
| 43 | \( 1 - 2.44iT - 43T^{2} \) |
| 47 | \( 1 + 2.92iT - 47T^{2} \) |
| 53 | \( 1 + 3.46T + 53T^{2} \) |
| 59 | \( 1 + 12.9iT - 59T^{2} \) |
| 61 | \( 1 + 4.89T + 61T^{2} \) |
| 67 | \( 1 + 5.46T + 67T^{2} \) |
| 71 | \( 1 - 6.92T + 71T^{2} \) |
| 73 | \( 1 + 0.656T + 73T^{2} \) |
| 79 | \( 1 + 3.86iT - 79T^{2} \) |
| 83 | \( 1 - 1.41T + 83T^{2} \) |
| 89 | \( 1 + 4.92iT - 89T^{2} \) |
| 97 | \( 1 - 4.92iT - 97T^{2} \) |
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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
−10.08335105344579291989134144786, −8.706784338157035549540419996627, −7.82348560382796709433859941141, −7.60604011643132907307695008000, −6.54278062482656326235443753699, −5.83596527940718132066496087348, −4.96859349686474719569485060844, −3.49972787533585385789511917688, −2.37894560704145831422315347377, −1.44476854162394081452273797114,
1.14805955310061395732974199869, 2.36331837421152666844938495890, 3.47552653823499652681979587580, 4.42261044718437761568336473139, 5.57681672188928364856654419671, 6.05727130164354052414904241740, 7.52577167342049437622318738885, 8.071289676527399896814547328704, 8.898405127629227581078996736293, 9.947084024417875847734536090614