L(s) = 1 | + (0.575 + 1.29i)2-s + (−0.669 + 0.743i)4-s + (0.913 + 0.406i)5-s + (0.294 − 1.38i)7-s + 1.41i·10-s + (1.95 − 0.415i)14-s + (0.104 + 0.994i)16-s + (−0.913 + 0.406i)20-s + (0.831 + 1.14i)28-s + (0.294 − 1.38i)29-s + (−0.104 + 0.994i)31-s + (−1.22 + 0.707i)32-s + (0.831 − 1.14i)35-s + (−0.309 − 0.951i)37-s + (0.669 + 0.743i)47-s + ⋯ |
L(s) = 1 | + (0.575 + 1.29i)2-s + (−0.669 + 0.743i)4-s + (0.913 + 0.406i)5-s + (0.294 − 1.38i)7-s + 1.41i·10-s + (1.95 − 0.415i)14-s + (0.104 + 0.994i)16-s + (−0.913 + 0.406i)20-s + (0.831 + 1.14i)28-s + (0.294 − 1.38i)29-s + (−0.104 + 0.994i)31-s + (−1.22 + 0.707i)32-s + (0.831 − 1.14i)35-s + (−0.309 − 0.951i)37-s + (0.669 + 0.743i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.125 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.125 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.167611856\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.167611856\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.575 - 1.29i)T + (-0.669 + 0.743i)T^{2} \) |
| 5 | \( 1 + (-0.913 - 0.406i)T + (0.669 + 0.743i)T^{2} \) |
| 7 | \( 1 + (-0.294 + 1.38i)T + (-0.913 - 0.406i)T^{2} \) |
| 13 | \( 1 + (0.978 + 0.207i)T^{2} \) |
| 17 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 19 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 23 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (-0.294 + 1.38i)T + (-0.913 - 0.406i)T^{2} \) |
| 31 | \( 1 + (0.104 - 0.994i)T + (-0.978 - 0.207i)T^{2} \) |
| 37 | \( 1 + (0.309 + 0.951i)T + (-0.809 + 0.587i)T^{2} \) |
| 41 | \( 1 + (-0.913 + 0.406i)T^{2} \) |
| 43 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (-0.669 - 0.743i)T + (-0.104 + 0.994i)T^{2} \) |
| 53 | \( 1 + (-0.809 - 0.587i)T + (0.309 + 0.951i)T^{2} \) |
| 59 | \( 1 + (0.669 - 0.743i)T + (-0.104 - 0.994i)T^{2} \) |
| 61 | \( 1 + (-1.40 + 0.147i)T + (0.978 - 0.207i)T^{2} \) |
| 67 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (0.809 - 0.587i)T + (0.309 - 0.951i)T^{2} \) |
| 73 | \( 1 + (1.34 - 0.437i)T + (0.809 - 0.587i)T^{2} \) |
| 79 | \( 1 + (-0.669 + 0.743i)T^{2} \) |
| 83 | \( 1 + (1.40 - 0.147i)T + (0.978 - 0.207i)T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + (-0.913 + 0.406i)T + (0.669 - 0.743i)T^{2} \) |
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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
−8.731821768285986639639712940882, −7.957139190088857614995302576863, −7.20941927590740452982160251642, −6.85213769652379214044711741396, −5.95955635257321804402052065884, −5.47684315144040900057435783660, −4.40012444900276874384104184513, −3.97762685274207666731829527502, −2.60739796545749032502474948626, −1.37607424651176604397270519754,
1.41781804331677058417518292933, 2.13326198831872801570686208993, 2.82653959918305657933545087216, 3.78836917822865543871087736945, 4.89718038686001289500829701962, 5.33008928341930879777842075473, 6.04338717224201977458064549008, 7.08105511442038114603536663168, 8.188962427769349898118132529978, 8.935152400133008447344172697323