L(s) = 1 | + (0.587 − 0.809i)3-s + (0.309 + 0.951i)5-s + 0.618i·7-s + (0.190 + 0.587i)13-s + (0.951 + 0.309i)15-s + (−0.951 − 1.30i)19-s + (0.500 + 0.363i)21-s + (0.951 + 0.309i)23-s + (−0.809 + 0.587i)25-s + (0.951 + 0.309i)27-s + (−1.30 − 0.951i)29-s + (−0.587 − 0.809i)31-s + (−0.587 + 0.190i)35-s + (−0.309 − 0.951i)37-s + (0.587 + 0.190i)39-s + ⋯ |
L(s) = 1 | + (0.587 − 0.809i)3-s + (0.309 + 0.951i)5-s + 0.618i·7-s + (0.190 + 0.587i)13-s + (0.951 + 0.309i)15-s + (−0.951 − 1.30i)19-s + (0.500 + 0.363i)21-s + (0.951 + 0.309i)23-s + (−0.809 + 0.587i)25-s + (0.951 + 0.309i)27-s + (−1.30 − 0.951i)29-s + (−0.587 − 0.809i)31-s + (−0.587 + 0.190i)35-s + (−0.309 − 0.951i)37-s + (0.587 + 0.190i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0941i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0941i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.217085292\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.217085292\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.309 - 0.951i)T \) |
good | 3 | \( 1 + (-0.587 + 0.809i)T + (-0.309 - 0.951i)T^{2} \) |
| 7 | \( 1 - 0.618iT - T^{2} \) |
| 11 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 13 | \( 1 + (-0.190 - 0.587i)T + (-0.809 + 0.587i)T^{2} \) |
| 17 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 19 | \( 1 + (0.951 + 1.30i)T + (-0.309 + 0.951i)T^{2} \) |
| 23 | \( 1 + (-0.951 - 0.309i)T + (0.809 + 0.587i)T^{2} \) |
| 29 | \( 1 + (1.30 + 0.951i)T + (0.309 + 0.951i)T^{2} \) |
| 31 | \( 1 + (0.587 + 0.809i)T + (-0.309 + 0.951i)T^{2} \) |
| 37 | \( 1 + (0.309 + 0.951i)T + (-0.809 + 0.587i)T^{2} \) |
| 41 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 43 | \( 1 + 1.61iT - T^{2} \) |
| 47 | \( 1 + (0.363 - 0.5i)T + (-0.309 - 0.951i)T^{2} \) |
| 53 | \( 1 + (-0.809 - 0.587i)T + (0.309 + 0.951i)T^{2} \) |
| 59 | \( 1 + (1.53 - 0.5i)T + (0.809 - 0.587i)T^{2} \) |
| 61 | \( 1 + (-0.309 + 0.951i)T + (-0.809 - 0.587i)T^{2} \) |
| 67 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 71 | \( 1 + (-0.363 + 0.5i)T + (-0.309 - 0.951i)T^{2} \) |
| 73 | \( 1 + (-0.309 + 0.951i)T + (-0.809 - 0.587i)T^{2} \) |
| 79 | \( 1 + (0.951 - 1.30i)T + (-0.309 - 0.951i)T^{2} \) |
| 83 | \( 1 + (-0.587 - 0.809i)T + (-0.309 + 0.951i)T^{2} \) |
| 89 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 97 | \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \) |
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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
−10.70008665261545129435102571691, −9.346156279804282727003416352525, −8.916106903831152728694037213943, −7.73452042945538029690784694539, −7.10803202294893138814817695815, −6.34426586490685552068014829226, −5.31622817989715229523872945707, −3.88833038792028134407616897199, −2.57856052430192880723067752895, −2.01518637056998855507874392963,
1.48160198417663858990829979556, 3.20307752340291635623029196937, 4.06290126220015714186479761468, 4.92057367600375059294397485505, 5.91850058211446209547036506391, 7.08352407393039407715291082121, 8.257766539249353292932676370082, 8.760581316940946647199114825535, 9.627997285487594393415620366078, 10.28198166550133449347622570320