L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.499 + 0.866i)4-s + (−0.5 − 0.866i)5-s − 0.999·6-s − 0.999·8-s + (−0.499 − 0.866i)9-s + (0.499 − 0.866i)10-s + (0.5 − 0.866i)11-s + (−0.499 − 0.866i)12-s − 13-s + 0.999·15-s + (−0.5 − 0.866i)16-s + (0.499 − 0.866i)18-s + (−1.5 − 2.59i)19-s + 0.999·20-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.288 + 0.499i)3-s + (−0.249 + 0.433i)4-s + (−0.223 − 0.387i)5-s − 0.408·6-s − 0.353·8-s + (−0.166 − 0.288i)9-s + (0.158 − 0.273i)10-s + (0.150 − 0.261i)11-s + (−0.144 − 0.249i)12-s − 0.277·13-s + 0.258·15-s + (−0.125 − 0.216i)16-s + (0.117 − 0.204i)18-s + (−0.344 − 0.596i)19-s + 0.223·20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8801287083\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8801287083\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 3 | \( 1 + (0.5 - 0.866i)T \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + (-0.5 + 0.866i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + T + 13T^{2} \) |
| 17 | \( 1 + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.5 + 2.59i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3.5 + 6.06i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 8T + 29T^{2} \) |
| 31 | \( 1 + (1 - 1.73i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (5.5 + 9.52i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 11T + 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 + (2.5 + 4.33i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-5.5 + 9.52i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-2 + 3.46i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 + (3 - 5.19i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-4 - 6.92i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 8T + 83T^{2} \) |
| 89 | \( 1 + (5 + 8.66i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 16T + 97T^{2} \) |
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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
−9.146353683229800724582652377772, −8.693723195080642662925728377664, −7.68998901340847150621717626859, −6.92262637012188349132149420093, −5.94326101734480387602690913637, −5.29671005939261916869100566876, −4.33181151033038606731622145041, −3.73272268281713611396952477324, −2.33887324136620453230471265959, −0.32403681845762107195321269734,
1.42972529726077457029203743100, 2.45322509333451396517058481957, 3.59067646456588672500919073783, 4.42344079111448365833187729963, 5.59649745273060555980122573302, 6.15990554813016074397789566380, 7.30966273343146199358405903726, 7.79346106087266620485001231542, 8.986722804677901127227257430761, 9.753454417185625800572825285905