| L(s) = 1 | + (1.88 − 0.284i)2-s + (−1.42 − 0.974i)3-s + (1.57 − 0.487i)4-s + (0.158 − 2.12i)5-s + (−2.98 − 1.43i)6-s + (−0.598 + 0.288i)8-s + (−0.00173 − 0.00441i)9-s + (−0.303 − 4.05i)10-s + (1.70 − 4.35i)11-s + (−2.73 − 0.842i)12-s + (0.609 + 0.764i)13-s + (−2.29 + 2.87i)15-s + (−3.77 + 2.57i)16-s + (−1.52 + 1.41i)17-s + (−0.00452 − 0.00784i)18-s + (2.64 − 4.58i)19-s + ⋯ |
| L(s) = 1 | + (1.33 − 0.201i)2-s + (−0.825 − 0.562i)3-s + (0.789 − 0.243i)4-s + (0.0710 − 0.948i)5-s + (−1.21 − 0.585i)6-s + (−0.211 + 0.101i)8-s + (−0.000577 − 0.00147i)9-s + (−0.0960 − 1.28i)10-s + (0.514 − 1.31i)11-s + (−0.788 − 0.243i)12-s + (0.169 + 0.211i)13-s + (−0.592 + 0.743i)15-s + (−0.944 + 0.644i)16-s + (−0.370 + 0.343i)17-s + (−0.00106 − 0.00184i)18-s + (0.606 − 1.05i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 343 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.221 + 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 343 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.221 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.16255 - 1.45559i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.16255 - 1.45559i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| good | 2 | \( 1 + (-1.88 + 0.284i)T + (1.91 - 0.589i)T^{2} \) |
| 3 | \( 1 + (1.42 + 0.974i)T + (1.09 + 2.79i)T^{2} \) |
| 5 | \( 1 + (-0.158 + 2.12i)T + (-4.94 - 0.745i)T^{2} \) |
| 11 | \( 1 + (-1.70 + 4.35i)T + (-8.06 - 7.48i)T^{2} \) |
| 13 | \( 1 + (-0.609 - 0.764i)T + (-2.89 + 12.6i)T^{2} \) |
| 17 | \( 1 + (1.52 - 1.41i)T + (1.27 - 16.9i)T^{2} \) |
| 19 | \( 1 + (-2.64 + 4.58i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.77 - 2.57i)T + (1.71 + 22.9i)T^{2} \) |
| 29 | \( 1 + (1.23 + 5.40i)T + (-26.1 + 12.5i)T^{2} \) |
| 31 | \( 1 + (-5.08 - 8.80i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1.65 + 0.508i)T + (30.5 + 20.8i)T^{2} \) |
| 41 | \( 1 + (-3.19 + 1.54i)T + (25.5 - 32.0i)T^{2} \) |
| 43 | \( 1 + (-9.05 - 4.36i)T + (26.8 + 33.6i)T^{2} \) |
| 47 | \( 1 + (-4.28 + 0.645i)T + (44.9 - 13.8i)T^{2} \) |
| 53 | \( 1 + (4.94 - 1.52i)T + (43.7 - 29.8i)T^{2} \) |
| 59 | \( 1 + (0.388 + 5.18i)T + (-58.3 + 8.79i)T^{2} \) |
| 61 | \( 1 + (-4.26 - 1.31i)T + (50.4 + 34.3i)T^{2} \) |
| 67 | \( 1 + (0.241 + 0.417i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (2.97 - 13.0i)T + (-63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (5.90 + 0.889i)T + (69.7 + 21.5i)T^{2} \) |
| 79 | \( 1 + (2.87 - 4.98i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-1.84 + 2.31i)T + (-18.4 - 80.9i)T^{2} \) |
| 89 | \( 1 + (4.12 + 10.5i)T + (-65.2 + 60.5i)T^{2} \) |
| 97 | \( 1 + 1.76T + 97T^{2} \) |
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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
−11.49430931399067144166277926934, −11.05026992163044640573063232983, −9.204767942848252864207269024336, −8.593160602853849233707591228967, −6.92862461539137249880724887079, −6.02358442483715055107176023364, −5.35532338081942716751931450335, −4.33655531705337936537025117519, −3.07382612515577888022524094463, −1.02013790934279413892510209526,
2.61931874996039975219249282882, 3.97537451224003798496584042365, 4.79911338121614975100557641362, 5.77477603883241714555553736609, 6.59966627146626737679495870904, 7.51568707442966578422351292471, 9.292783771451134011289130341398, 10.22701377575613109150822189006, 11.03984398104142903961214215465, 11.90412372511536499131479740065
