L(s) = 1 | + (0.781 + 0.623i)3-s + (−0.433 + 0.900i)4-s + (−0.222 + 0.974i)7-s + (0.222 + 0.974i)9-s + (−0.900 + 0.433i)12-s + (−0.781 + 0.623i)13-s + (−0.623 − 0.781i)16-s + (−0.752 − 0.752i)19-s + (−0.781 + 0.623i)21-s + (0.974 − 0.222i)25-s + (−0.433 + 0.900i)27-s + (−0.781 − 0.623i)28-s + (0.467 + 0.467i)31-s + (−0.974 − 0.222i)36-s + (−0.623 − 0.218i)37-s + ⋯ |
L(s) = 1 | + (0.781 + 0.623i)3-s + (−0.433 + 0.900i)4-s + (−0.222 + 0.974i)7-s + (0.222 + 0.974i)9-s + (−0.900 + 0.433i)12-s + (−0.781 + 0.623i)13-s + (−0.623 − 0.781i)16-s + (−0.752 − 0.752i)19-s + (−0.781 + 0.623i)21-s + (0.974 − 0.222i)25-s + (−0.433 + 0.900i)27-s + (−0.781 − 0.623i)28-s + (0.467 + 0.467i)31-s + (−0.974 − 0.222i)36-s + (−0.623 − 0.218i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.699 - 0.714i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1911 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.699 - 0.714i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.158664046\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.158664046\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.781 - 0.623i)T \) |
| 7 | \( 1 + (0.222 - 0.974i)T \) |
| 13 | \( 1 + (0.781 - 0.623i)T \) |
good | 2 | \( 1 + (0.433 - 0.900i)T^{2} \) |
| 5 | \( 1 + (-0.974 + 0.222i)T^{2} \) |
| 11 | \( 1 + (-0.433 + 0.900i)T^{2} \) |
| 17 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 19 | \( 1 + (0.752 + 0.752i)T + iT^{2} \) |
| 23 | \( 1 + (0.623 + 0.781i)T^{2} \) |
| 29 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 31 | \( 1 + (-0.467 - 0.467i)T + iT^{2} \) |
| 37 | \( 1 + (0.623 + 0.218i)T + (0.781 + 0.623i)T^{2} \) |
| 41 | \( 1 + (-0.974 + 0.222i)T^{2} \) |
| 43 | \( 1 + (-1.52 + 1.21i)T + (0.222 - 0.974i)T^{2} \) |
| 47 | \( 1 + (0.433 - 0.900i)T^{2} \) |
| 53 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 59 | \( 1 + (-0.974 - 0.222i)T^{2} \) |
| 61 | \( 1 + (-0.781 - 1.62i)T + (-0.623 + 0.781i)T^{2} \) |
| 67 | \( 1 + (1.40 - 1.40i)T - iT^{2} \) |
| 71 | \( 1 + (-0.781 + 0.623i)T^{2} \) |
| 73 | \( 1 + (-1.59 - 1.00i)T + (0.433 + 0.900i)T^{2} \) |
| 79 | \( 1 - 0.867T + T^{2} \) |
| 83 | \( 1 + (0.433 + 0.900i)T^{2} \) |
| 89 | \( 1 + (-0.433 - 0.900i)T^{2} \) |
| 97 | \( 1 + (1.19 + 1.19i)T + iT^{2} \) |
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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
−9.407059978521123725085279430929, −8.782272944049519226379783295244, −8.531259487931218510964465432193, −7.43276067914280515363953006504, −6.78114807063798735388563258539, −5.41204432485811736976638591705, −4.64286709682309904332336952440, −3.92284777812872334972558066976, −2.81757080639271810877273738238, −2.32876963012563875367070219757,
0.77808179657141195119759123039, 1.94182407377703039712639458764, 3.13395538438290038394050678857, 4.13446321485738701684030728796, 4.96448990361670762190252600429, 6.12631290226378496079318157622, 6.71729147950052290709095701885, 7.64356799291405925119086145011, 8.228503917272585843120718369171, 9.222062597649733693281600023175