| L(s) = 1 | + i·3-s + i·5-s + 1.70·7-s − 9-s + 3.79·11-s + 1.61·13-s − 15-s + 2.05i·17-s − 0.518·19-s + 1.70i·21-s + (−4.69 − 0.988i)23-s − 25-s − i·27-s − 2.88·29-s + 4.16i·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + 0.447i·5-s + 0.643·7-s − 0.333·9-s + 1.14·11-s + 0.448·13-s − 0.258·15-s + 0.498i·17-s − 0.119·19-s + 0.371i·21-s + (−0.978 − 0.206i)23-s − 0.200·25-s − 0.192i·27-s − 0.535·29-s + 0.748i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.310 - 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.310 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.045020184\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.045020184\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (4.69 + 0.988i)T \) |
| good | 7 | \( 1 - 1.70T + 7T^{2} \) |
| 11 | \( 1 - 3.79T + 11T^{2} \) |
| 13 | \( 1 - 1.61T + 13T^{2} \) |
| 17 | \( 1 - 2.05iT - 17T^{2} \) |
| 19 | \( 1 + 0.518T + 19T^{2} \) |
| 29 | \( 1 + 2.88T + 29T^{2} \) |
| 31 | \( 1 - 4.16iT - 31T^{2} \) |
| 37 | \( 1 - 7.56iT - 37T^{2} \) |
| 41 | \( 1 - 11.8T + 41T^{2} \) |
| 43 | \( 1 + 4.43T + 43T^{2} \) |
| 47 | \( 1 - 0.599iT - 47T^{2} \) |
| 53 | \( 1 + 2.97iT - 53T^{2} \) |
| 59 | \( 1 - 9.67iT - 59T^{2} \) |
| 61 | \( 1 + 2.26iT - 61T^{2} \) |
| 67 | \( 1 - 9.80T + 67T^{2} \) |
| 71 | \( 1 - 9.51iT - 71T^{2} \) |
| 73 | \( 1 + 7.31T + 73T^{2} \) |
| 79 | \( 1 - 7.33T + 79T^{2} \) |
| 83 | \( 1 - 4.63T + 83T^{2} \) |
| 89 | \( 1 - 16.0iT - 89T^{2} \) |
| 97 | \( 1 + 14.4iT - 97T^{2} \) |
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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
−8.391816309532432933125597094769, −7.79426528662253225138580073006, −6.81411488818707607463921740110, −6.23840875556294053002987353769, −5.52581926154309936954592614402, −4.55231290218120486901089577493, −3.97845864457296593250648723984, −3.26689122278431770145987491542, −2.14694920599887478936678827622, −1.20374872156259294409560338940,
0.56840435121473957590523487527, 1.57179600562302942149221833564, 2.26409217237648921067088437034, 3.57138402560501626586761117882, 4.20585416783454357756129667770, 5.06523025023818978027453056757, 5.93718603647348774969931176917, 6.40006247756074555422595987488, 7.40947284150059640010862886821, 7.83793109686308366454087525600
