L(s) = 1 | + (−1.14 − 1.14i)2-s + (0.891 + 0.453i)3-s + 1.61i·4-s + (−0.5 − 1.53i)6-s + (−0.831 + 0.831i)7-s + (0.707 − 0.707i)8-s + (0.587 + 0.809i)9-s + (−0.734 + 1.44i)12-s + 1.90·14-s + (0.437 + 0.437i)17-s + (0.253 − 1.59i)18-s + (−1.11 + 0.363i)21-s + (0.951 − 0.309i)24-s + (0.156 + 0.987i)27-s + (−1.34 − 1.34i)28-s + ⋯ |
L(s) = 1 | + (−1.14 − 1.14i)2-s + (0.891 + 0.453i)3-s + 1.61i·4-s + (−0.5 − 1.53i)6-s + (−0.831 + 0.831i)7-s + (0.707 − 0.707i)8-s + (0.587 + 0.809i)9-s + (−0.734 + 1.44i)12-s + 1.90·14-s + (0.437 + 0.437i)17-s + (0.253 − 1.59i)18-s + (−1.11 + 0.363i)21-s + (0.951 − 0.309i)24-s + (0.156 + 0.987i)27-s + (−1.34 − 1.34i)28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.646 - 0.762i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.646 - 0.762i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7038812806\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7038812806\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.891 - 0.453i)T \) |
| 5 | \( 1 \) |
| 47 | \( 1 + (0.707 + 0.707i)T \) |
good | 2 | \( 1 + (1.14 + 1.14i)T + iT^{2} \) |
| 7 | \( 1 + (0.831 - 0.831i)T - iT^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 13 | \( 1 - iT^{2} \) |
| 17 | \( 1 + (-0.437 - 0.437i)T + iT^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + iT^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (1.34 - 1.34i)T - iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 53 | \( 1 + (1.14 - 1.14i)T - iT^{2} \) |
| 59 | \( 1 + 1.17T + T^{2} \) |
| 61 | \( 1 - 1.61T + T^{2} \) |
| 67 | \( 1 + iT^{2} \) |
| 71 | \( 1 - 1.90iT - T^{2} \) |
| 73 | \( 1 - iT^{2} \) |
| 79 | \( 1 + 0.618iT - T^{2} \) |
| 83 | \( 1 + (-1.41 + 1.41i)T - iT^{2} \) |
| 89 | \( 1 + 1.17T + T^{2} \) |
| 97 | \( 1 + (-0.831 + 0.831i)T - iT^{2} \) |
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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
−9.055461225363784963947207893762, −8.411977360482367832301301010398, −7.88363473372293466016823706180, −6.90107401393402898643933306942, −5.87500659415461524054239603875, −4.84786439833948612220057906017, −3.63536878720010620694743144247, −3.14286833708921301787017889595, −2.37920531494847129494509772922, −1.48650650610145134318395422319,
0.54659724276299136973108284035, 1.75539062138956730014176721173, 3.13694949539534630183784631043, 3.81244308330635455807384296371, 5.09565852315987065636392034291, 6.16419519217095767232399288629, 6.74317500351549417355638532783, 7.29455531480306789669206184782, 7.83865627327475553884011074297, 8.527030813426590398098859094708