| L(s) = 1 | + (−0.234 + 0.536i)2-s + (−1.27 + 1.17i)3-s + (1.12 + 1.21i)4-s + (−1.79 − 0.269i)5-s + (−0.334 − 0.957i)6-s + (2.59 + 2.40i)7-s + (−2.02 + 0.707i)8-s + (0.225 − 2.99i)9-s + (0.564 − 0.898i)10-s + (−0.974 + 0.838i)11-s + (−2.86 − 0.215i)12-s + (−2.06 + 3.02i)13-s + (−1.89 + 0.828i)14-s + (2.59 − 1.76i)15-s + (−0.154 + 2.05i)16-s + (−2.82 − 2.82i)17-s + ⋯ |
| L(s) = 1 | + (−0.165 + 0.379i)2-s + (−0.733 + 0.679i)3-s + (0.563 + 0.607i)4-s + (−0.801 − 0.120i)5-s + (−0.136 − 0.390i)6-s + (0.980 + 0.909i)7-s + (−0.714 + 0.250i)8-s + (0.0752 − 0.997i)9-s + (0.178 − 0.283i)10-s + (−0.293 + 0.252i)11-s + (−0.826 − 0.0621i)12-s + (−0.571 + 0.838i)13-s + (−0.507 + 0.221i)14-s + (0.669 − 0.456i)15-s + (−0.0385 + 0.513i)16-s + (−0.684 − 0.684i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.929 - 0.368i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.929 - 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.147081 + 0.771116i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.147081 + 0.771116i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.27 - 1.17i)T \) |
| 29 | \( 1 + (1.28 - 5.23i)T \) |
| good | 2 | \( 1 + (0.234 - 0.536i)T + (-1.36 - 1.46i)T^{2} \) |
| 5 | \( 1 + (1.79 + 0.269i)T + (4.77 + 1.47i)T^{2} \) |
| 7 | \( 1 + (-2.59 - 2.40i)T + (0.523 + 6.98i)T^{2} \) |
| 11 | \( 1 + (0.974 - 0.838i)T + (1.63 - 10.8i)T^{2} \) |
| 13 | \( 1 + (2.06 - 3.02i)T + (-4.74 - 12.1i)T^{2} \) |
| 17 | \( 1 + (2.82 + 2.82i)T + 17iT^{2} \) |
| 19 | \( 1 + (5.81 + 3.65i)T + (8.24 + 17.1i)T^{2} \) |
| 23 | \( 1 + (-8.33 - 3.27i)T + (16.8 + 15.6i)T^{2} \) |
| 31 | \( 1 + (-3.72 + 2.74i)T + (9.13 - 29.6i)T^{2} \) |
| 37 | \( 1 + (-2.62 - 7.49i)T + (-28.9 + 23.0i)T^{2} \) |
| 41 | \( 1 + (-6.95 - 1.86i)T + (35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (-1.32 + 1.79i)T + (-12.6 - 41.0i)T^{2} \) |
| 47 | \( 1 + (-1.08 - 1.26i)T + (-7.00 + 46.4i)T^{2} \) |
| 53 | \( 1 + (-6.55 - 5.22i)T + (11.7 + 51.6i)T^{2} \) |
| 59 | \( 1 + (7.47 - 4.31i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-13.5 + 0.506i)T + (60.8 - 4.55i)T^{2} \) |
| 67 | \( 1 + (-10.9 + 0.819i)T + (66.2 - 9.98i)T^{2} \) |
| 71 | \( 1 + (3.89 - 1.87i)T + (44.2 - 55.5i)T^{2} \) |
| 73 | \( 1 + (-0.963 + 8.55i)T + (-71.1 - 16.2i)T^{2} \) |
| 79 | \( 1 + (-0.335 + 1.77i)T + (-73.5 - 28.8i)T^{2} \) |
| 83 | \( 1 + (0.302 - 0.980i)T + (-68.5 - 46.7i)T^{2} \) |
| 89 | \( 1 + (2.15 - 0.242i)T + (86.7 - 19.8i)T^{2} \) |
| 97 | \( 1 + (0.817 + 1.54i)T + (-54.6 + 80.1i)T^{2} \) |
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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
−11.99765212591360249588545360064, −11.46946064434968153256332837814, −10.94072792563807607621451041919, −9.251632813132901303823704813069, −8.609721260005858404931029280654, −7.41872313239674267594784390235, −6.54260482425345292420371964112, −5.12863478772002369810442191930, −4.31127203550797160037291547683, −2.58891434033142035697080879119,
0.69096279217020225538143137256, 2.27307466641504418426603724157, 4.25040619474159515914671422724, 5.49730019719160103496423602931, 6.65512302626549142370160792103, 7.56351146080921854858517302192, 8.357519393612497433064573199023, 10.20559349525106568395890140543, 10.97127094190882966852468836910, 11.18441922195657589821509005759
