L(s) = 1 | − i·3-s − i·5-s + 3.60·7-s − 9-s + 5.78·11-s + 0.896·13-s − 15-s − 6.83i·17-s + 1.90·19-s − 3.60i·21-s + (4.77 + 0.469i)23-s − 25-s + i·27-s + 1.39·29-s − 1.12i·31-s + ⋯ |
L(s) = 1 | − 0.577i·3-s − 0.447i·5-s + 1.36·7-s − 0.333·9-s + 1.74·11-s + 0.248·13-s − 0.258·15-s − 1.65i·17-s + 0.437·19-s − 0.786i·21-s + (0.995 + 0.0979i)23-s − 0.200·25-s + 0.192i·27-s + 0.258·29-s − 0.201i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.412 + 0.910i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.412 + 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.942731456\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.942731456\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + iT \) |
| 23 | \( 1 + (-4.77 - 0.469i)T \) |
good | 7 | \( 1 - 3.60T + 7T^{2} \) |
| 11 | \( 1 - 5.78T + 11T^{2} \) |
| 13 | \( 1 - 0.896T + 13T^{2} \) |
| 17 | \( 1 + 6.83iT - 17T^{2} \) |
| 19 | \( 1 - 1.90T + 19T^{2} \) |
| 29 | \( 1 - 1.39T + 29T^{2} \) |
| 31 | \( 1 + 1.12iT - 31T^{2} \) |
| 37 | \( 1 - 7.48iT - 37T^{2} \) |
| 41 | \( 1 - 3.19T + 41T^{2} \) |
| 43 | \( 1 - 2.53T + 43T^{2} \) |
| 47 | \( 1 - 5.80iT - 47T^{2} \) |
| 53 | \( 1 + 2.67iT - 53T^{2} \) |
| 59 | \( 1 + 6.58iT - 59T^{2} \) |
| 61 | \( 1 + 0.597iT - 61T^{2} \) |
| 67 | \( 1 + 5.41T + 67T^{2} \) |
| 71 | \( 1 - 3.72iT - 71T^{2} \) |
| 73 | \( 1 - 0.479T + 73T^{2} \) |
| 79 | \( 1 - 1.03T + 79T^{2} \) |
| 83 | \( 1 + 11.2T + 83T^{2} \) |
| 89 | \( 1 - 5.21iT - 89T^{2} \) |
| 97 | \( 1 - 4.73iT - 97T^{2} \) |
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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
−7.998677608022436464176303829856, −7.31347033918756087218523397801, −6.73027231621130647383155205261, −5.89830011971009374426026851633, −4.96596009129024935402678600368, −4.58657814860979708698840088002, −3.53668131336284750710051369447, −2.51583847932343612525625115619, −1.37330383764224492986473880074, −0.985721750412085283415943800007,
1.17987770131923945184722872937, 1.90220848144671956100112896535, 3.14768971299892843977038661074, 4.04248034498270004473170931738, 4.37282363126987665011855504218, 5.44063650031965121199934860303, 6.05543127058581457776522736945, 6.86878537449959142554734220420, 7.58523741407424736096713661172, 8.471760132515962081182484289361