L(s) = 1 | + (−0.809 + 0.587i)4-s + (−1.13 − 1.56i)7-s + (−0.492 + 0.159i)13-s + (0.309 − 0.951i)16-s + (−0.831 + 1.14i)19-s + (0.809 + 0.587i)25-s + (1.83 + 0.596i)28-s + (0.535 + 1.64i)31-s + 1.41i·43-s + (−0.844 + 2.59i)49-s + (0.304 − 0.418i)52-s + (1.34 + 0.437i)61-s + (0.309 + 0.951i)64-s − 1.73·67-s + (−0.304 − 0.418i)73-s + ⋯ |
L(s) = 1 | + (−0.809 + 0.587i)4-s + (−1.13 − 1.56i)7-s + (−0.492 + 0.159i)13-s + (0.309 − 0.951i)16-s + (−0.831 + 1.14i)19-s + (0.809 + 0.587i)25-s + (1.83 + 0.596i)28-s + (0.535 + 1.64i)31-s + 1.41i·43-s + (−0.844 + 2.59i)49-s + (0.304 − 0.418i)52-s + (1.34 + 0.437i)61-s + (0.309 + 0.951i)64-s − 1.73·67-s + (−0.304 − 0.418i)73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.105 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3267 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.105 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5084928248\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5084928248\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 5 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 7 | \( 1 + (1.13 + 1.56i)T + (-0.309 + 0.951i)T^{2} \) |
| 13 | \( 1 + (0.492 - 0.159i)T + (0.809 - 0.587i)T^{2} \) |
| 17 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 19 | \( 1 + (0.831 - 1.14i)T + (-0.309 - 0.951i)T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 31 | \( 1 + (-0.535 - 1.64i)T + (-0.809 + 0.587i)T^{2} \) |
| 37 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 41 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 43 | \( 1 - 1.41iT - T^{2} \) |
| 47 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 53 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 59 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 61 | \( 1 + (-1.34 - 0.437i)T + (0.809 + 0.587i)T^{2} \) |
| 67 | \( 1 + 1.73T + T^{2} \) |
| 71 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 73 | \( 1 + (0.304 + 0.418i)T + (-0.309 + 0.951i)T^{2} \) |
| 79 | \( 1 + (-0.492 + 0.159i)T + (0.809 - 0.587i)T^{2} \) |
| 83 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + (-0.309 - 0.951i)T + (-0.809 + 0.587i)T^{2} \) |
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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
−9.051231326211761942788019023430, −8.257182242482379676395163267013, −7.50389787019201477383881240553, −6.90681732541223864727579472436, −6.18244061513063455055842729299, −4.97212523801421561005704019331, −4.27904039296474071752662941925, −3.58348915984394351229030868609, −2.91714526215407673227707678162, −1.17067295121211329376219747123,
0.34669964667033080075886463480, 2.19168363399318001586107407332, 2.85759782762158150729223672888, 4.02893505616840435110266225467, 4.90896047274438158454242020589, 5.61301910866459448418082226058, 6.25339330649354929453520665190, 6.91281092020145091713910841683, 8.172122468793051809769186250228, 8.848465226937882216976576090273