L(s) = 1 | + (−0.5 − 1.32i)2-s − 2.64i·3-s + (−1.50 + 1.32i)4-s + (−3.50 + 1.32i)6-s − 4·7-s + (2.50 + 1.32i)8-s − 4.00·9-s − 2.64i·11-s + (3.50 + 3.96i)12-s + (2 + 5.29i)14-s + (0.500 − 3.96i)16-s + 3·17-s + (2.00 + 5.29i)18-s − 2.64i·19-s + 10.5i·21-s + (−3.50 + 1.32i)22-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.935i)2-s − 1.52i·3-s + (−0.750 + 0.661i)4-s + (−1.42 + 0.540i)6-s − 1.51·7-s + (0.883 + 0.467i)8-s − 1.33·9-s − 0.797i·11-s + (1.01 + 1.14i)12-s + (0.534 + 1.41i)14-s + (0.125 − 0.992i)16-s + 0.727·17-s + (0.471 + 1.24i)18-s − 0.606i·19-s + 2.30i·21-s + (−0.746 + 0.282i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.883 - 0.467i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.883 - 0.467i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.152453 + 0.614068i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.152453 + 0.614068i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 1.32i)T \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + 2.64iT - 3T^{2} \) |
| 7 | \( 1 + 4T + 7T^{2} \) |
| 11 | \( 1 + 2.64iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 3T + 17T^{2} \) |
| 19 | \( 1 + 2.64iT - 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + 10.5iT - 37T^{2} \) |
| 41 | \( 1 + 5T + 41T^{2} \) |
| 43 | \( 1 + 5.29iT - 43T^{2} \) |
| 47 | \( 1 - 8T + 47T^{2} \) |
| 53 | \( 1 - 10.5iT - 53T^{2} \) |
| 59 | \( 1 + 5.29iT - 59T^{2} \) |
| 61 | \( 1 + 10.5iT - 61T^{2} \) |
| 67 | \( 1 - 7.93iT - 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 - 7T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 - 7.93iT - 83T^{2} \) |
| 89 | \( 1 + T + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−12.27663322469443509525454591162, −11.08447538029037148799030712718, −9.948425133667698485563105980070, −8.937468376288715202210951689621, −7.893476631881274387086974573659, −6.88185987515442619539598265197, −5.79093089168755001397541133467, −3.58908890321402914729492351441, −2.43013874299533290114921261546, −0.62036380948190494547338357103,
3.45433537376310144986067606263, 4.54698847704968478810073318200, 5.73191656978605985197092450600, 6.75150824391477632329555764682, 8.148204716509848239891592748303, 9.347067482071268746374524742045, 9.959362729867815666331769878362, 10.34197819489543942038129863563, 12.05585048460620626423862543456, 13.23784010218841081582418256214