| L(s) = 1 | + (0.255 + 1.16i)2-s + (−2.31 + 1.90i)3-s + (2.34 − 1.08i)4-s + (2.94 + 0.818i)5-s + (−2.80 − 2.20i)6-s + (7.54 − 7.14i)7-s + (4.74 + 6.23i)8-s + (1.74 − 8.83i)9-s + (−0.197 + 3.63i)10-s + (−3.02 + 2.56i)11-s + (−3.36 + 6.98i)12-s + (6.78 − 2.28i)13-s + (10.2 + 6.93i)14-s + (−8.39 + 3.72i)15-s + (0.655 − 0.772i)16-s + (−10.1 + 10.7i)17-s + ⋯ |
| L(s) = 1 | + (0.127 + 0.581i)2-s + (−0.772 + 0.635i)3-s + (0.586 − 0.271i)4-s + (0.589 + 0.163i)5-s + (−0.467 − 0.367i)6-s + (1.07 − 1.02i)7-s + (0.592 + 0.779i)8-s + (0.193 − 0.981i)9-s + (−0.0197 + 0.363i)10-s + (−0.274 + 0.233i)11-s + (−0.280 + 0.581i)12-s + (0.521 − 0.175i)13-s + (0.730 + 0.495i)14-s + (−0.559 + 0.248i)15-s + (0.0409 − 0.0482i)16-s + (−0.596 + 0.629i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.571 - 0.820i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.571 - 0.820i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.59215 + 0.831638i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.59215 + 0.831638i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (2.31 - 1.90i)T \) |
| 59 | \( 1 + (0.524 + 58.9i)T \) |
| good | 2 | \( 1 + (-0.255 - 1.16i)T + (-3.63 + 1.67i)T^{2} \) |
| 5 | \( 1 + (-2.94 - 0.818i)T + (21.4 + 12.8i)T^{2} \) |
| 7 | \( 1 + (-7.54 + 7.14i)T + (2.65 - 48.9i)T^{2} \) |
| 11 | \( 1 + (3.02 - 2.56i)T + (19.5 - 119. i)T^{2} \) |
| 13 | \( 1 + (-6.78 + 2.28i)T + (134. - 102. i)T^{2} \) |
| 17 | \( 1 + (10.1 - 10.7i)T + (-15.6 - 288. i)T^{2} \) |
| 19 | \( 1 + (-12.6 - 31.8i)T + (-262. + 248. i)T^{2} \) |
| 23 | \( 1 + (2.41 + 22.2i)T + (-516. + 113. i)T^{2} \) |
| 29 | \( 1 + (7.08 - 32.1i)T + (-763. - 353. i)T^{2} \) |
| 31 | \( 1 + (-5.89 + 14.8i)T + (-697. - 660. i)T^{2} \) |
| 37 | \( 1 + (25.1 + 19.1i)T + (366. + 1.31e3i)T^{2} \) |
| 41 | \( 1 + (-7.06 + 64.9i)T + (-1.64e3 - 361. i)T^{2} \) |
| 43 | \( 1 + (30.6 - 36.0i)T + (-299. - 1.82e3i)T^{2} \) |
| 47 | \( 1 + (-40.9 + 11.3i)T + (1.89e3 - 1.13e3i)T^{2} \) |
| 53 | \( 1 + (38.8 - 2.10i)T + (2.79e3 - 303. i)T^{2} \) |
| 61 | \( 1 + (-4.38 + 0.964i)T + (3.37e3 - 1.56e3i)T^{2} \) |
| 67 | \( 1 + (22.7 - 17.2i)T + (1.20e3 - 4.32e3i)T^{2} \) |
| 71 | \( 1 + (-76.2 + 21.1i)T + (4.31e3 - 2.59e3i)T^{2} \) |
| 73 | \( 1 + (64.0 - 94.4i)T + (-1.97e3 - 4.95e3i)T^{2} \) |
| 79 | \( 1 + (2.68 + 16.3i)T + (-5.91e3 + 1.99e3i)T^{2} \) |
| 83 | \( 1 + (121. - 64.5i)T + (3.86e3 - 5.70e3i)T^{2} \) |
| 89 | \( 1 + (-11.1 + 50.8i)T + (-7.18e3 - 3.32e3i)T^{2} \) |
| 97 | \( 1 + (89.7 + 132. i)T + (-3.48e3 + 8.74e3i)T^{2} \) |
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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
−12.50320361727861478721382747064, −11.30162652768237782003640441325, −10.61049988324061124509199642559, −10.07388022222757704516835481104, −8.324458331245490159769508186194, −7.20498183647858527612992463082, −6.11953957400336483010218445821, −5.27493099666140655463653174265, −4.04240014555560261120572906651, −1.61332759215369573207247243896,
1.51212723910678240899679161161, 2.60743058288199408088316729624, 4.83196596264917520366426755773, 5.82417931638679334042913491808, 7.00845148537413012292701245992, 8.091873465925723179801756379774, 9.395252874382045424706043427004, 10.82243592097797521030135808286, 11.58478669106984997463823074981, 11.88058131744707794031336276302
