L(s) = 1 | + (0.258 − 0.965i)2-s + (−0.866 − 0.499i)4-s + (1.12 − 1.46i)5-s + (−0.707 + 0.707i)8-s + (−0.965 − 0.258i)9-s + (−1.12 − 1.46i)10-s + (0.500 + 0.866i)16-s + (−0.965 + 0.258i)17-s + (−0.499 + 0.866i)18-s + (−1.70 + 0.707i)20-s + (−0.624 − 2.33i)25-s + (−0.707 − 1.70i)29-s + (0.965 − 0.258i)32-s + i·34-s + (0.707 + 0.707i)36-s + (−1.46 − 1.12i)37-s + ⋯ |
L(s) = 1 | + (0.258 − 0.965i)2-s + (−0.866 − 0.499i)4-s + (1.12 − 1.46i)5-s + (−0.707 + 0.707i)8-s + (−0.965 − 0.258i)9-s + (−1.12 − 1.46i)10-s + (0.500 + 0.866i)16-s + (−0.965 + 0.258i)17-s + (−0.499 + 0.866i)18-s + (−1.70 + 0.707i)20-s + (−0.624 − 2.33i)25-s + (−0.707 − 1.70i)29-s + (0.965 − 0.258i)32-s + i·34-s + (0.707 + 0.707i)36-s + (−1.46 − 1.12i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.972 - 0.233i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.972 - 0.233i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.142440303\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.142440303\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.258 + 0.965i)T \) |
| 7 | \( 1 \) |
| 17 | \( 1 + (0.965 - 0.258i)T \) |
good | 3 | \( 1 + (0.965 + 0.258i)T^{2} \) |
| 5 | \( 1 + (-1.12 + 1.46i)T + (-0.258 - 0.965i)T^{2} \) |
| 11 | \( 1 + (-0.258 + 0.965i)T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 19 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 23 | \( 1 + (0.965 - 0.258i)T^{2} \) |
| 29 | \( 1 + (0.707 + 1.70i)T + (-0.707 + 0.707i)T^{2} \) |
| 31 | \( 1 + (0.965 + 0.258i)T^{2} \) |
| 37 | \( 1 + (1.46 + 1.12i)T + (0.258 + 0.965i)T^{2} \) |
| 41 | \( 1 + (0.292 - 0.707i)T + (-0.707 - 0.707i)T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2} \) |
| 59 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 61 | \( 1 + (-0.758 + 0.0999i)T + (0.965 - 0.258i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 73 | \( 1 + (-1.83 - 0.241i)T + (0.965 + 0.258i)T^{2} \) |
| 79 | \( 1 + (0.965 - 0.258i)T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 + (1.73 - i)T + (0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + (-0.707 - 1.70i)T + (-0.707 + 0.707i)T^{2} \) |
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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
−8.605806865736493017659059497601, −8.189322803013431255270351322054, −6.60694446302236595815407365992, −5.71905655997961533965453186555, −5.41002486286836278034573531181, −4.50990933550106358646456071087, −3.74648874306236670675154747446, −2.43011682657594202268918753013, −1.87578948984808099513278947038, −0.58539248629331278947778383125,
2.04036288934781870613153549600, 2.96917741610359589758718964647, 3.62955722604908126287703879846, 5.01896021867004421173504562248, 5.53836384695720570934189535091, 6.29800337338599488824301582589, 6.90454733560109414127536949830, 7.36326093813315675667726618535, 8.550003453549906757520847215807, 8.968604085769015501170310569408