L(s) = 1 | + (−0.5 + 0.866i)2-s + (0.866 − 0.5i)3-s + (−0.499 − 0.866i)4-s + (0.668 − 2.13i)5-s + 0.999i·6-s + (2.13 − 1.23i)7-s + 0.999·8-s + (0.499 − 0.866i)9-s + (1.51 + 1.64i)10-s − 0.452·11-s + (−0.866 − 0.499i)12-s + (−1.54 − 2.67i)13-s + 2.46i·14-s + (−0.487 − 2.18i)15-s + (−0.5 + 0.866i)16-s + (1.05 − 1.83i)17-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (0.499 − 0.288i)3-s + (−0.249 − 0.433i)4-s + (0.299 − 0.954i)5-s + 0.408i·6-s + (0.806 − 0.465i)7-s + 0.353·8-s + (0.166 − 0.288i)9-s + (0.478 + 0.520i)10-s − 0.136·11-s + (−0.249 − 0.144i)12-s + (−0.428 − 0.741i)13-s + 0.658i·14-s + (−0.125 − 0.563i)15-s + (−0.125 + 0.216i)16-s + (0.256 − 0.444i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.429 + 0.902i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.429 + 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.659084029\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.659084029\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 3 | \( 1 + (-0.866 + 0.5i)T \) |
| 5 | \( 1 + (-0.668 + 2.13i)T \) |
| 37 | \( 1 + (-0.0294 + 6.08i)T \) |
good | 7 | \( 1 + (-2.13 + 1.23i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + 0.452T + 11T^{2} \) |
| 13 | \( 1 + (1.54 + 2.67i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-1.05 + 1.83i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.58 + 1.49i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + 1.77T + 23T^{2} \) |
| 29 | \( 1 - 8.36iT - 29T^{2} \) |
| 31 | \( 1 - 2.55iT - 31T^{2} \) |
| 41 | \( 1 + (1.18 + 2.04i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + 3.94T + 43T^{2} \) |
| 47 | \( 1 - 2.64iT - 47T^{2} \) |
| 53 | \( 1 + (8.26 + 4.77i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (2.78 + 1.60i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-7.10 + 4.10i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-8.26 + 4.77i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-0.00323 - 0.00560i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 9.13iT - 73T^{2} \) |
| 79 | \( 1 + (-10.9 + 6.33i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-6.55 - 3.78i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-9.02 - 5.21i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 5.64T + 97T^{2} \) |
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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
−9.442145025493923323960942420612, −8.806224111299524772855210574426, −7.914563396960184985617679171961, −7.55254712689416204888378045700, −6.46825899899182348111557390963, −5.20414844231660591313645086754, −4.89559733290734298132700242731, −3.46892525552248710637112120951, −1.92744378839868396887928538429, −0.798600097696909228068744461261,
1.77040342609595792449643543813, 2.53354224783844172266249950698, 3.58646199212691386457246089218, 4.58214906622256742431066861398, 5.69900023167303361039577849031, 6.78600537225782425011684262055, 7.84294881123293079400502270673, 8.291864910030503694083786069152, 9.474172405886996278234660976928, 9.882288628819549263727468199490