L(s) = 1 | + i·3-s + (−0.471 − 2.18i)5-s + 1.05i·7-s − 9-s − 5.95·11-s − 5.75i·13-s + (2.18 − 0.471i)15-s + 4.60i·17-s + 4.92·19-s − 1.05·21-s − 5.41i·23-s + (−4.55 + 2.06i)25-s − i·27-s − 2.23·29-s − 3.73·31-s + ⋯ |
L(s) = 1 | + 0.577i·3-s + (−0.211 − 0.977i)5-s + 0.398i·7-s − 0.333·9-s − 1.79·11-s − 1.59i·13-s + (0.564 − 0.121i)15-s + 1.11i·17-s + 1.13·19-s − 0.230·21-s − 1.12i·23-s + (−0.910 + 0.412i)25-s − 0.192i·27-s − 0.415·29-s − 0.670·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.211 - 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.211 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7806714691\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7806714691\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + (0.471 + 2.18i)T \) |
| 67 | \( 1 - iT \) |
good | 7 | \( 1 - 1.05iT - 7T^{2} \) |
| 11 | \( 1 + 5.95T + 11T^{2} \) |
| 13 | \( 1 + 5.75iT - 13T^{2} \) |
| 17 | \( 1 - 4.60iT - 17T^{2} \) |
| 19 | \( 1 - 4.92T + 19T^{2} \) |
| 23 | \( 1 + 5.41iT - 23T^{2} \) |
| 29 | \( 1 + 2.23T + 29T^{2} \) |
| 31 | \( 1 + 3.73T + 31T^{2} \) |
| 37 | \( 1 - 10.8iT - 37T^{2} \) |
| 41 | \( 1 - 12.0T + 41T^{2} \) |
| 43 | \( 1 + 6.81iT - 43T^{2} \) |
| 47 | \( 1 - 8.17iT - 47T^{2} \) |
| 53 | \( 1 - 1.55iT - 53T^{2} \) |
| 59 | \( 1 + 14.2T + 59T^{2} \) |
| 61 | \( 1 - 2.79T + 61T^{2} \) |
| 71 | \( 1 + 14.7T + 71T^{2} \) |
| 73 | \( 1 - 7.57iT - 73T^{2} \) |
| 79 | \( 1 - 11.8T + 79T^{2} \) |
| 83 | \( 1 - 17.4iT - 83T^{2} \) |
| 89 | \( 1 - 1.38T + 89T^{2} \) |
| 97 | \( 1 - 2.06iT - 97T^{2} \) |
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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
−8.612040079639179874552191866769, −7.83101034502525693929710111090, −7.71978566400882023231133283375, −6.10045579686454009048103720918, −5.46966303055369614873947182275, −5.08193403437783015287447677015, −4.19272870926742563917990064892, −3.16857122041584645541656729909, −2.46965134683655231174551980178, −0.964176813992616286973271055101,
0.25773321124527547176626827905, 1.83656075246468042866381571611, 2.65230567709251323940753188054, 3.41228097497390170948720538564, 4.42371075792677178927074530022, 5.40092288559423373939209645799, 6.03104745238556012082034926569, 7.14643697228983124629732990371, 7.42813976750217551092273821690, 7.77106300122895904148157744194