| L(s) = 1 | − i·2-s − i·3-s − 4-s + (1.38 − 1.75i)5-s − 6-s − 2.50i·7-s + i·8-s − 9-s + (−1.75 − 1.38i)10-s − 3.77·11-s + i·12-s + 1.36i·13-s − 2.50·14-s + (−1.75 − 1.38i)15-s + 16-s − 0.221i·17-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.577i·3-s − 0.5·4-s + (0.621 − 0.783i)5-s − 0.408·6-s − 0.946i·7-s + 0.353i·8-s − 0.333·9-s + (−0.554 − 0.439i)10-s − 1.13·11-s + 0.288i·12-s + 0.378i·13-s − 0.669·14-s + (−0.452 − 0.358i)15-s + 0.250·16-s − 0.0538i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.783 - 0.621i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.783 - 0.621i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9739814367\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9739814367\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 + (-1.38 + 1.75i)T \) |
| 37 | \( 1 + iT \) |
| good | 7 | \( 1 + 2.50iT - 7T^{2} \) |
| 11 | \( 1 + 3.77T + 11T^{2} \) |
| 13 | \( 1 - 1.36iT - 13T^{2} \) |
| 17 | \( 1 + 0.221iT - 17T^{2} \) |
| 19 | \( 1 + 6.14T + 19T^{2} \) |
| 23 | \( 1 + 0.867iT - 23T^{2} \) |
| 29 | \( 1 + 0.646T + 29T^{2} \) |
| 31 | \( 1 - 3.86T + 31T^{2} \) |
| 41 | \( 1 + 0.910T + 41T^{2} \) |
| 43 | \( 1 + 4.59iT - 43T^{2} \) |
| 47 | \( 1 - 0.778iT - 47T^{2} \) |
| 53 | \( 1 - 2.27iT - 53T^{2} \) |
| 59 | \( 1 + 7.78T + 59T^{2} \) |
| 61 | \( 1 - 9.42T + 61T^{2} \) |
| 67 | \( 1 + 10.7iT - 67T^{2} \) |
| 71 | \( 1 + 11.2T + 71T^{2} \) |
| 73 | \( 1 - 2.86iT - 73T^{2} \) |
| 79 | \( 1 - 0.495T + 79T^{2} \) |
| 83 | \( 1 - 1.08iT - 83T^{2} \) |
| 89 | \( 1 - 0.0899T + 89T^{2} \) |
| 97 | \( 1 + 4.46iT - 97T^{2} \) |
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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
−9.410246369497354024708398162560, −8.544544858099810472867362072700, −7.889170671541137479711215860597, −6.84255414912758054777955553623, −5.86434091754368699202409558453, −4.86995317005912798588626364534, −4.08357744590493360597047920361, −2.64724121788004805401375214292, −1.70105532780846435828617846507, −0.40680560265482172922703555516,
2.27154281280887708935293397561, 3.12850766507139320573022124673, 4.48488230063239785398104685869, 5.47216442027899941546624759816, 5.98035792733947000083510444214, 6.88376598506639873943752599288, 7.974654848983528472084661454549, 8.631800044543804752930135137984, 9.516682486352178386914862649745, 10.27947387983122328043184774469
