| L(s) = 1 | + (−0.766 + 0.642i)2-s + (−0.939 − 0.342i)3-s + (0.173 − 0.984i)4-s + (−0.445 − 2.52i)5-s + (0.939 − 0.342i)6-s + (0.5 − 0.866i)7-s + (0.500 + 0.866i)8-s + (0.766 + 0.642i)9-s + (1.96 + 1.64i)10-s + (1.61 + 2.80i)11-s + (−0.499 + 0.866i)12-s + (0.978 − 0.356i)13-s + (0.173 + 0.984i)14-s + (−0.445 + 2.52i)15-s + (−0.939 − 0.342i)16-s + (2.18 − 1.83i)17-s + ⋯ |
| L(s) = 1 | + (−0.541 + 0.454i)2-s + (−0.542 − 0.197i)3-s + (0.0868 − 0.492i)4-s + (−0.199 − 1.12i)5-s + (0.383 − 0.139i)6-s + (0.188 − 0.327i)7-s + (0.176 + 0.306i)8-s + (0.255 + 0.214i)9-s + (0.620 + 0.520i)10-s + (0.488 + 0.845i)11-s + (−0.144 + 0.250i)12-s + (0.271 − 0.0988i)13-s + (0.0464 + 0.263i)14-s + (−0.114 + 0.651i)15-s + (−0.234 − 0.0855i)16-s + (0.529 − 0.444i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.159 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.159 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.658153 - 0.560331i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.658153 - 0.560331i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.766 - 0.642i)T \) |
| 3 | \( 1 + (0.939 + 0.342i)T \) |
| 7 | \( 1 + (-0.5 + 0.866i)T \) |
| 19 | \( 1 + (-3.33 + 2.80i)T \) |
| good | 5 | \( 1 + (0.445 + 2.52i)T + (-4.69 + 1.71i)T^{2} \) |
| 11 | \( 1 + (-1.61 - 2.80i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.978 + 0.356i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-2.18 + 1.83i)T + (2.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (0.790 - 4.48i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (5.24 + 4.40i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-3.84 + 6.65i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 4.63T + 37T^{2} \) |
| 41 | \( 1 + (11.4 + 4.15i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (1.84 + 10.4i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (-3.25 - 2.72i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (-0.0500 + 0.284i)T + (-49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (-0.251 + 0.211i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (-0.222 + 1.26i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-0.0419 - 0.0351i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (2.56 + 14.5i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (-13.8 - 5.03i)T + (55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (-9.01 - 3.27i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (4.21 - 7.30i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-7.75 + 2.82i)T + (68.1 - 57.2i)T^{2} \) |
| 97 | \( 1 + (0.638 - 0.536i)T + (16.8 - 95.5i)T^{2} \) |
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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
−9.796777896550005759503722785605, −9.321563848717670560588983593124, −8.287740417635780098277927079371, −7.50849934352828184913636036222, −6.79428190534287262395713727159, −5.55008277480938617668004462932, −4.96410773938263970948731281090, −3.87286459449514577184764379524, −1.78531770863073467352179111412, −0.61742609076097742054919070479,
1.41246467825636539070736245336, 3.04347152659115455500294168701, 3.69798783289729307515324990704, 5.17053780253637787998016493749, 6.27673256932103607835954128141, 6.92588794935554150511557205294, 8.042443729365188001224218644621, 8.787259533943771765090733265898, 9.874112570738342173911794524427, 10.54794432896355812200333698842
