| L(s) = 1 | + (−0.786 − 0.618i)2-s + (−0.0197 + 0.0277i)3-s + (0.235 + 0.971i)4-s + (3.71 − 0.715i)5-s + (0.0326 − 0.00959i)6-s + (0.765 + 2.53i)7-s + (0.415 − 0.909i)8-s + (0.980 + 2.83i)9-s + (−3.36 − 1.73i)10-s + (−4.60 + 3.62i)11-s + (−0.0316 − 0.0126i)12-s + (1.18 − 0.763i)13-s + (0.963 − 2.46i)14-s + (−0.0535 + 0.117i)15-s + (−0.888 + 0.458i)16-s + (−2.31 − 2.20i)17-s + ⋯ |
| L(s) = 1 | + (−0.555 − 0.437i)2-s + (−0.0114 + 0.0160i)3-s + (0.117 + 0.485i)4-s + (1.66 − 0.320i)5-s + (0.0133 − 0.00391i)6-s + (0.289 + 0.957i)7-s + (0.146 − 0.321i)8-s + (0.326 + 0.944i)9-s + (−1.06 − 0.548i)10-s + (−1.38 + 1.09i)11-s + (−0.00913 − 0.00365i)12-s + (0.329 − 0.211i)13-s + (0.257 − 0.658i)14-s + (−0.0138 + 0.0302i)15-s + (−0.222 + 0.114i)16-s + (−0.561 − 0.535i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.188i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.981 - 0.188i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.27526 + 0.121592i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.27526 + 0.121592i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.786 + 0.618i)T \) |
| 7 | \( 1 + (-0.765 - 2.53i)T \) |
| 23 | \( 1 + (2.45 - 4.12i)T \) |
| good | 3 | \( 1 + (0.0197 - 0.0277i)T + (-0.981 - 2.83i)T^{2} \) |
| 5 | \( 1 + (-3.71 + 0.715i)T + (4.64 - 1.85i)T^{2} \) |
| 11 | \( 1 + (4.60 - 3.62i)T + (2.59 - 10.6i)T^{2} \) |
| 13 | \( 1 + (-1.18 + 0.763i)T + (5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (2.31 + 2.20i)T + (0.808 + 16.9i)T^{2} \) |
| 19 | \( 1 + (-1.93 + 1.84i)T + (0.904 - 18.9i)T^{2} \) |
| 29 | \( 1 + (-4.98 + 1.46i)T + (24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (-6.89 + 0.658i)T + (30.4 - 5.86i)T^{2} \) |
| 37 | \( 1 + (-0.106 - 0.306i)T + (-29.0 + 22.8i)T^{2} \) |
| 41 | \( 1 + (6.54 + 7.55i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (4.88 + 10.6i)T + (-28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 + (-4.21 + 7.29i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.387 + 8.14i)T + (-52.7 - 5.03i)T^{2} \) |
| 59 | \( 1 + (4.40 + 2.27i)T + (34.2 + 48.0i)T^{2} \) |
| 61 | \( 1 + (-1.64 - 2.31i)T + (-19.9 + 57.6i)T^{2} \) |
| 67 | \( 1 + (-2.00 + 0.802i)T + (48.4 - 46.2i)T^{2} \) |
| 71 | \( 1 + (0.860 - 5.98i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (-1.46 - 6.02i)T + (-64.8 + 33.4i)T^{2} \) |
| 79 | \( 1 + (-0.189 - 3.97i)T + (-78.6 + 7.50i)T^{2} \) |
| 83 | \( 1 + (-5.37 + 6.20i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (14.3 + 1.36i)T + (87.3 + 16.8i)T^{2} \) |
| 97 | \( 1 + (-6.57 - 7.59i)T + (-13.8 + 96.0i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.59596011730021677670380012959, −10.19030624857537605206061993626, −10.12958233853766383372287868537, −8.972950985548140632920220490688, −8.117684330358023682169227347107, −6.91845423685876815148858632875, −5.45170452816700766276148107670, −4.94789075017256286943051769355, −2.50327648932154131044982634663, −1.95276087924924569940875818294,
1.25684064986650497987329020309, 2.92530014178961101494811212216, 4.76505059081709143909844747303, 6.15566290229127781269912565556, 6.44297099194894988686442243563, 7.86638330108963707601640018742, 8.771849930001872280717358091245, 9.966141618744129981234916222443, 10.31264746154606719397464280241, 11.18403129292567776436494622125
