L(s) = 1 | + (−0.5 − 0.866i)3-s + (0.866 − 0.5i)5-s + (−0.5 + 0.866i)9-s + (−0.866 − 0.5i)15-s − 17-s + (−0.866 − 0.5i)19-s + 23-s + (0.5 − 0.866i)25-s + 27-s + (0.5 − 0.866i)29-s + (0.866 + 0.5i)31-s + i·37-s + (0.866 + 0.5i)41-s + (0.5 + 0.866i)43-s + i·45-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.866i)3-s + (0.866 − 0.5i)5-s + (−0.5 + 0.866i)9-s + (−0.866 − 0.5i)15-s − 17-s + (−0.866 − 0.5i)19-s + 23-s + (0.5 − 0.866i)25-s + 27-s + (0.5 − 0.866i)29-s + (0.866 + 0.5i)31-s + i·37-s + (0.866 + 0.5i)41-s + (0.5 + 0.866i)43-s + i·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.362 - 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.362 - 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.352716815 - 0.9251832715i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.352716815 - 0.9251832715i\) |
\(L(1)\) |
\(\approx\) |
\(0.9903917112 - 0.3613967989i\) |
\(L(1)\) |
\(\approx\) |
\(0.9903917112 - 0.3613967989i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
| 13 | \( 1 \) |
good | 3 | \( 1 + (-0.5 - 0.866i)T \) |
| 5 | \( 1 + (0.866 - 0.5i)T \) |
| 17 | \( 1 - T \) |
| 19 | \( 1 + (-0.866 - 0.5i)T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + (0.5 - 0.866i)T \) |
| 31 | \( 1 + (0.866 + 0.5i)T \) |
| 37 | \( 1 + iT \) |
| 41 | \( 1 + (0.866 + 0.5i)T \) |
| 43 | \( 1 + (0.5 + 0.866i)T \) |
| 47 | \( 1 + (0.866 - 0.5i)T \) |
| 53 | \( 1 + (-0.5 + 0.866i)T \) |
| 59 | \( 1 - iT \) |
| 61 | \( 1 + (-0.5 + 0.866i)T \) |
| 67 | \( 1 + (0.866 - 0.5i)T \) |
| 71 | \( 1 + (0.866 - 0.5i)T \) |
| 73 | \( 1 + (-0.866 - 0.5i)T \) |
| 79 | \( 1 + (-0.5 - 0.866i)T \) |
| 83 | \( 1 - iT \) |
| 89 | \( 1 - iT \) |
| 97 | \( 1 + (0.866 - 0.5i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−18.55832126118981624660961015056, −17.56645278791402698679899809838, −17.410569374233468298120377540214, −16.701459876465571766313430721876, −15.7568431788527009088960030850, −15.38386949848053819521107929415, −14.37619122771528006179306943436, −14.191960818235446721729035657793, −13.012316047574037104153509469686, −12.56191444032850638169051563428, −11.48721081913106729393410397367, −10.8282761934571207057920286087, −10.510820767580695986178232900996, −9.63281289206898975341641070769, −9.082116047066832461362365495986, −8.40757864746899878972932157477, −7.14786231589056728357563331867, −6.52628883274722510176374401855, −5.894555249666893350405975794, −5.17660344422194114404616066831, −4.42052416746686014920538320870, −3.64542386887082968319187612145, −2.72483629002135807850364033734, −2.01763570216952324556467734544, −0.76607538680573210381623302708,
0.68829238591249292687803350547, 1.41992635378039250649082775834, 2.35822989286047655559424544468, 2.83571649403995739210005033225, 4.504117497653394821521106388234, 4.76323195394150800885764662731, 5.85514691003080340147884915517, 6.32678662918551239107344768145, 6.93589590672202681726014675819, 7.8637556084785111154134259459, 8.66558393420505965350556330684, 9.1540784030094041594518057472, 10.18999737016097601848826855652, 10.811182379135713072917751256658, 11.53024940203861505330102419691, 12.3170936948089412644202061749, 12.95200481353512436019869213976, 13.46137817165254255253136153968, 13.95293338822259913666644178573, 14.93257054048021571505682773417, 15.709428008998383162675172395752, 16.55731454417066506007737126230, 17.29235014105567483569820771767, 17.46537433396373165943114289228, 18.24687264462162777838685187367