L(s) = 1 | + (0.584 − 1.28i)2-s + 2.81i·3-s + (−1.31 − 1.50i)4-s + (0.390 + 2.20i)5-s + (3.62 + 1.64i)6-s − 4.13·7-s + (−2.70 + 0.813i)8-s − 4.91·9-s + (3.06 + 0.785i)10-s − 0.681i·11-s + (4.23 − 3.70i)12-s + 3.19·13-s + (−2.41 + 5.31i)14-s + (−6.19 + 1.09i)15-s + (−0.535 + 3.96i)16-s + 2.93i·17-s + ⋯ |
L(s) = 1 | + (0.413 − 0.910i)2-s + 1.62i·3-s + (−0.658 − 0.752i)4-s + (0.174 + 0.984i)5-s + (1.47 + 0.671i)6-s − 1.56·7-s + (−0.957 + 0.287i)8-s − 1.63·9-s + (0.968 + 0.248i)10-s − 0.205i·11-s + (1.22 − 1.06i)12-s + 0.886·13-s + (−0.645 + 1.42i)14-s + (−1.59 + 0.283i)15-s + (−0.133 + 0.990i)16-s + 0.711i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.348 - 0.937i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.348 - 0.937i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.563719 + 0.810875i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.563719 + 0.810875i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.584 + 1.28i)T \) |
| 5 | \( 1 + (-0.390 - 2.20i)T \) |
| 19 | \( 1 + (2.23 - 3.74i)T \) |
good | 3 | \( 1 - 2.81iT - 3T^{2} \) |
| 7 | \( 1 + 4.13T + 7T^{2} \) |
| 11 | \( 1 + 0.681iT - 11T^{2} \) |
| 13 | \( 1 - 3.19T + 13T^{2} \) |
| 17 | \( 1 - 2.93iT - 17T^{2} \) |
| 23 | \( 1 + 6.41T + 23T^{2} \) |
| 29 | \( 1 + 0.928iT - 29T^{2} \) |
| 31 | \( 1 - 6.68T + 31T^{2} \) |
| 37 | \( 1 - 6.63T + 37T^{2} \) |
| 41 | \( 1 - 4.89iT - 41T^{2} \) |
| 43 | \( 1 - 7.02T + 43T^{2} \) |
| 47 | \( 1 - 6.74T + 47T^{2} \) |
| 53 | \( 1 + 7.60T + 53T^{2} \) |
| 59 | \( 1 - 13.1T + 59T^{2} \) |
| 61 | \( 1 + 5.76T + 61T^{2} \) |
| 67 | \( 1 - 5.12iT - 67T^{2} \) |
| 71 | \( 1 - 7.43T + 71T^{2} \) |
| 73 | \( 1 + 10.1iT - 73T^{2} \) |
| 79 | \( 1 + 4.94T + 79T^{2} \) |
| 83 | \( 1 + 9.74T + 83T^{2} \) |
| 89 | \( 1 - 7.65iT - 89T^{2} \) |
| 97 | \( 1 + 15.0T + 97T^{2} \) |
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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
−11.29500864818359836274741161664, −10.55779610929089452345988884365, −9.999043778728675216006204650740, −9.518927724627628974744830854011, −8.367029616940049688807981009093, −6.19254273213257235863646370377, −5.94287877696971602381502222910, −4.14296912842832078420200028907, −3.64447625538307447053721398745, −2.69142089814349695827834934085,
0.56463624632307741525955634092, 2.63930489720638251480989812145, 4.17328205427687164218359526391, 5.73405632581647868305533044526, 6.32785558654709613022814230633, 7.09661120327993258464048243473, 8.094768783421270383555023749127, 8.891402930489147472518746716598, 9.757844595853997534699485470435, 11.63562970728003318187087444299