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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.857 + 0.513i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.857 + 0.513i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4990970136\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4990970136\) |
\(L(1)\) |
|
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
−8.536669785340520284805266917819, −7.78311233166723214210977444409, −7.11982054697682443500380691766, −6.06926002977903501667606926976, −5.49583275483239784029617103143, −4.23103649958547817164715455917, −3.59203465735480828869873169455, −2.99578771417037214733723937268, −2.18898294719655216801026385420, −0.30305642801264828773008794294,
1.22132719365775948707904406000, 3.17178971763624815474324823581, 3.80155519659434301809077941568, 4.78562120760175674211069047496, 5.24612929555661203764497400753, 5.90858042797506718771179198779, 7.07010383448935827280508226175, 7.71281745122318492633701398624, 8.370018213772099215843859413378, 8.751012327664801136550873018747