| L(s) = 1 | + (0.797 + 1.09i)2-s + (0.000161 − 0.00301i)3-s + (0.0511 − 0.159i)4-s + (−1.71 + 0.0366i)5-s + (0.00342 − 0.00222i)6-s + (1.78 + 1.95i)7-s + (2.78 − 0.924i)8-s + (2.98 + 0.319i)9-s + (−1.40 − 1.84i)10-s + (0.141 − 1.19i)11-s + (−0.000472 − 0.000179i)12-s + (0.608 + 1.16i)13-s + (−0.716 + 3.50i)14-s + (−0.000165 + 0.00516i)15-s + (2.96 + 2.11i)16-s + (−0.0180 + 0.00432i)17-s + ⋯ |
| L(s) = 1 | + (0.564 + 0.773i)2-s + (9.30e−5 − 0.00173i)3-s + (0.0255 − 0.0798i)4-s + (−0.765 + 0.0163i)5-s + (0.00139 − 0.000909i)6-s + (0.673 + 0.739i)7-s + (0.984 − 0.326i)8-s + (0.994 + 0.106i)9-s + (−0.444 − 0.582i)10-s + (0.0426 − 0.360i)11-s + (−0.000136 − 5.19e−5i)12-s + (0.168 + 0.323i)13-s + (−0.191 + 0.937i)14-s + (−4.27e−5 + 0.00133i)15-s + (0.740 + 0.528i)16-s + (−0.00437 + 0.00104i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 343 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.606 - 0.794i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 343 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.606 - 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.69468 + 0.838352i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.69468 + 0.838352i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-1.78 - 1.95i)T \) |
| good | 2 | \( 1 + (-0.797 - 1.09i)T + (-0.609 + 1.90i)T^{2} \) |
| 3 | \( 1 + (-0.000161 + 0.00301i)T + (-2.98 - 0.319i)T^{2} \) |
| 5 | \( 1 + (1.71 - 0.0366i)T + (4.99 - 0.213i)T^{2} \) |
| 11 | \( 1 + (-0.141 + 1.19i)T + (-10.6 - 2.56i)T^{2} \) |
| 13 | \( 1 + (-0.608 - 1.16i)T + (-7.43 + 10.6i)T^{2} \) |
| 17 | \( 1 + (0.0180 - 0.00432i)T + (15.1 - 7.70i)T^{2} \) |
| 19 | \( 1 + (2.73 - 4.73i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.36 + 1.06i)T + (5.59 - 22.3i)T^{2} \) |
| 29 | \( 1 + (-2.50 + 1.01i)T + (20.8 - 20.1i)T^{2} \) |
| 31 | \( 1 + (-2.35 + 2.18i)T + (2.31 - 30.9i)T^{2} \) |
| 37 | \( 1 + (2.35 + 2.47i)T + (-1.97 + 36.9i)T^{2} \) |
| 41 | \( 1 + (-2.11 + 5.75i)T + (-31.2 - 26.5i)T^{2} \) |
| 43 | \( 1 + (8.47 + 2.81i)T + (34.4 + 25.7i)T^{2} \) |
| 47 | \( 1 + (4.60 + 2.46i)T + (26.0 + 39.1i)T^{2} \) |
| 53 | \( 1 + (4.16 + 2.46i)T + (25.5 + 46.4i)T^{2} \) |
| 59 | \( 1 + (-3.53 + 4.24i)T + (-10.6 - 58.0i)T^{2} \) |
| 61 | \( 1 + (7.00 - 0.600i)T + (60.1 - 10.3i)T^{2} \) |
| 67 | \( 1 + (4.32 - 1.33i)T + (55.3 - 37.7i)T^{2} \) |
| 71 | \( 1 + (-2.33 - 0.944i)T + (51.0 + 49.3i)T^{2} \) |
| 73 | \( 1 + (2.94 - 1.57i)T + (40.4 - 60.7i)T^{2} \) |
| 79 | \( 1 + (8.79 - 1.32i)T + (75.4 - 23.2i)T^{2} \) |
| 83 | \( 1 + (3.27 - 0.638i)T + (76.9 - 31.1i)T^{2} \) |
| 89 | \( 1 + (1.79 - 1.81i)T + (-0.951 - 88.9i)T^{2} \) |
| 97 | \( 1 + (0.116 + 0.511i)T + (-87.3 + 42.0i)T^{2} \) |
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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
−11.75259000452228731209408261500, −10.80434311389724988370365809103, −9.891813687639508875207189179636, −8.501849583581218552229851646712, −7.76616686521500435204999080304, −6.78082200972948487387979505915, −5.79880584643296820253650369995, −4.72896835579774482806773860166, −3.87378672725633240087645553445, −1.75778672534229341434143774157,
1.49647771772771724134059937018, 3.15885858970030552047958124784, 4.31831775076710686907434150039, 4.75716024150674334301536896127, 6.80258909185576313611056926989, 7.58534351365785530571192356924, 8.380680534199957694474833052400, 9.905629237613168463894976976300, 10.75909262138927130309948321191, 11.43871695320157826208273730309
