| 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
−7.77106300122895904148157744194, −7.42813976750217551092273821690, −7.14643697228983124629732990371, −6.03104745238556012082034926569, −5.40092288559423373939209645799, −4.42371075792677178927074530022, −3.41228097497390170948720538564, −2.65230567709251323940753188054, −1.83656075246468042866381571611, −0.25773321124527547176626827905,
0.964176813992616286973271055101, 2.46965134683655231174551980178, 3.16857122041584645541656729909, 4.19272870926742563917990064892, 5.08193403437783015287447677015, 5.46966303055369614873947182275, 6.10045579686454009048103720918, 7.71978566400882023231133283375, 7.83101034502525693929710111090, 8.612040079639179874552191866769
