L(s) = 1 | + (−0.451 + 0.892i)2-s + (0.754 + 0.656i)3-s + (−0.592 − 0.805i)4-s + (−0.926 + 0.376i)6-s + (0.401 + 0.915i)7-s + (0.986 − 0.164i)8-s + (0.137 + 0.990i)9-s + (0.789 + 0.614i)11-s + (0.0825 − 0.996i)12-s + (0.191 − 0.981i)13-s + (−0.998 − 0.0550i)14-s + (−0.298 + 0.954i)16-s + (0.592 − 0.805i)17-s + (−0.945 − 0.324i)18-s + (−0.298 + 0.954i)21-s + (−0.904 + 0.426i)22-s + ⋯ |
L(s) = 1 | + (−0.451 + 0.892i)2-s + (0.754 + 0.656i)3-s + (−0.592 − 0.805i)4-s + (−0.926 + 0.376i)6-s + (0.401 + 0.915i)7-s + (0.986 − 0.164i)8-s + (0.137 + 0.990i)9-s + (0.789 + 0.614i)11-s + (0.0825 − 0.996i)12-s + (0.191 − 0.981i)13-s + (−0.998 − 0.0550i)14-s + (−0.298 + 0.954i)16-s + (0.592 − 0.805i)17-s + (−0.945 − 0.324i)18-s + (−0.298 + 0.954i)21-s + (−0.904 + 0.426i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.315 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.315 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.145494756 + 1.587264593i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.145494756 + 1.587264593i\) |
\(L(1)\) |
\(\approx\) |
\(0.9858239033 + 0.7611733390i\) |
\(L(1)\) |
\(\approx\) |
\(0.9858239033 + 0.7611733390i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + (-0.451 + 0.892i)T \) |
| 3 | \( 1 + (0.754 + 0.656i)T \) |
| 7 | \( 1 + (0.401 + 0.915i)T \) |
| 11 | \( 1 + (0.789 + 0.614i)T \) |
| 13 | \( 1 + (0.191 - 0.981i)T \) |
| 17 | \( 1 + (0.592 - 0.805i)T \) |
| 23 | \( 1 + (0.754 - 0.656i)T \) |
| 29 | \( 1 + (0.993 + 0.110i)T \) |
| 31 | \( 1 + (0.546 - 0.837i)T \) |
| 37 | \( 1 + (-0.789 - 0.614i)T \) |
| 41 | \( 1 + (0.0275 + 0.999i)T \) |
| 43 | \( 1 + (-0.716 + 0.697i)T \) |
| 47 | \( 1 + (0.926 - 0.376i)T \) |
| 53 | \( 1 + (0.926 - 0.376i)T \) |
| 59 | \( 1 + (0.0275 + 0.999i)T \) |
| 61 | \( 1 + (0.635 - 0.771i)T \) |
| 67 | \( 1 + (-0.350 + 0.936i)T \) |
| 71 | \( 1 + (0.635 + 0.771i)T \) |
| 73 | \( 1 + (0.592 - 0.805i)T \) |
| 79 | \( 1 + (0.716 - 0.697i)T \) |
| 83 | \( 1 + (0.677 - 0.735i)T \) |
| 89 | \( 1 + (-0.592 - 0.805i)T \) |
| 97 | \( 1 + (-0.350 - 0.936i)T \) |
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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
−19.71956150976509233326251719265, −19.335873872822591006125799274182, −18.8224242965989146960815063394, −17.87827222213448284611104276730, −17.14319632672489864725266480423, −16.72942958971189929500685835143, −15.49026875832991256753148701213, −14.320755810013316322208102527445, −13.89296149126233105508511756698, −13.42369794907829333073215015726, −12.30329448180899057515298325090, −11.86966383155272734961993115698, −10.94875364931786085215845621012, −10.21597427866011502861198249501, −9.28752272074208929944681482154, −8.630671065310430018565817531437, −8.04839116608425717016642126293, −7.100983491228306451050103906121, −6.52970061120786547910797807855, −5.01851503244841294414831722409, −3.774133135280177764234958044383, −3.66468968891966167066001883146, −2.43066774829052779659221633601, −1.40384540210264942806475783169, −1.019645860049266028549704666532,
1.05384281468258537131787221960, 2.223725860750707419111558153706, 3.14864175548873949658000613649, 4.3382885889585607616094799578, 5.004494959426379926438424152418, 5.71256373871807716441139632469, 6.770366059059836336801869648658, 7.648182338281986195895273379356, 8.38800504381224884423523396577, 8.91469805132526266661337133068, 9.70253831697174139125578763569, 10.26333314318913968245550141496, 11.25128951714462599178173663819, 12.2676953298273828737314296931, 13.2550337017844978984647259239, 14.08576410746144313340668593314, 14.83249366580335108614258967354, 15.115562153182076696685110718226, 15.9061700217036984515394689240, 16.58092933928425680795624287348, 17.451619194609459102665215426064, 18.15844377334378884136210303771, 18.86925952081501752156990802965, 19.59927591391811198265987907203, 20.33407822053384967750122967834