| L(s) = 1 | + (−0.712 − 3.23i)2-s + (−2.95 + 0.530i)3-s + (−6.33 + 2.93i)4-s + (9.05 + 2.51i)5-s + (3.81 + 9.17i)6-s + (−1.39 + 1.32i)7-s + (5.98 + 7.87i)8-s + (8.43 − 3.13i)9-s + (1.68 − 31.0i)10-s + (9.09 − 7.72i)11-s + (17.1 − 12.0i)12-s + (7.09 − 2.38i)13-s + (5.26 + 3.57i)14-s + (−28.0 − 2.62i)15-s + (3.13 − 3.69i)16-s + (−4.87 + 5.14i)17-s + ⋯ |
| L(s) = 1 | + (−0.356 − 1.61i)2-s + (−0.984 + 0.176i)3-s + (−1.58 + 0.733i)4-s + (1.81 + 0.502i)5-s + (0.636 + 1.52i)6-s + (−0.199 + 0.188i)7-s + (0.748 + 0.984i)8-s + (0.937 − 0.347i)9-s + (0.168 − 3.10i)10-s + (0.827 − 0.702i)11-s + (1.43 − 1.00i)12-s + (0.545 − 0.183i)13-s + (0.376 + 0.255i)14-s + (−1.87 − 0.174i)15-s + (0.196 − 0.230i)16-s + (−0.286 + 0.302i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 + 0.963i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.266 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.711137 - 0.934260i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.711137 - 0.934260i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (2.95 - 0.530i)T \) |
| 59 | \( 1 + (-44.6 - 38.5i)T \) |
| good | 2 | \( 1 + (0.712 + 3.23i)T + (-3.63 + 1.67i)T^{2} \) |
| 5 | \( 1 + (-9.05 - 2.51i)T + (21.4 + 12.8i)T^{2} \) |
| 7 | \( 1 + (1.39 - 1.32i)T + (2.65 - 48.9i)T^{2} \) |
| 11 | \( 1 + (-9.09 + 7.72i)T + (19.5 - 119. i)T^{2} \) |
| 13 | \( 1 + (-7.09 + 2.38i)T + (134. - 102. i)T^{2} \) |
| 17 | \( 1 + (4.87 - 5.14i)T + (-15.6 - 288. i)T^{2} \) |
| 19 | \( 1 + (-5.67 - 14.2i)T + (-262. + 248. i)T^{2} \) |
| 23 | \( 1 + (3.05 + 28.0i)T + (-516. + 113. i)T^{2} \) |
| 29 | \( 1 + (-6.69 + 30.4i)T + (-763. - 353. i)T^{2} \) |
| 31 | \( 1 + (-22.0 + 55.4i)T + (-697. - 660. i)T^{2} \) |
| 37 | \( 1 + (4.54 + 3.45i)T + (366. + 1.31e3i)T^{2} \) |
| 41 | \( 1 + (0.643 - 5.91i)T + (-1.64e3 - 361. i)T^{2} \) |
| 43 | \( 1 + (-5.19 + 6.11i)T + (-299. - 1.82e3i)T^{2} \) |
| 47 | \( 1 + (36.2 - 10.0i)T + (1.89e3 - 1.13e3i)T^{2} \) |
| 53 | \( 1 + (-49.2 + 2.67i)T + (2.79e3 - 303. i)T^{2} \) |
| 61 | \( 1 + (-99.7 + 21.9i)T + (3.37e3 - 1.56e3i)T^{2} \) |
| 67 | \( 1 + (50.5 - 38.4i)T + (1.20e3 - 4.32e3i)T^{2} \) |
| 71 | \( 1 + (119. - 33.1i)T + (4.31e3 - 2.59e3i)T^{2} \) |
| 73 | \( 1 + (56.8 - 83.8i)T + (-1.97e3 - 4.95e3i)T^{2} \) |
| 79 | \( 1 + (2.70 + 16.5i)T + (-5.91e3 + 1.99e3i)T^{2} \) |
| 83 | \( 1 + (-30.2 + 16.0i)T + (3.86e3 - 5.70e3i)T^{2} \) |
| 89 | \( 1 + (7.07 - 32.1i)T + (-7.18e3 - 3.32e3i)T^{2} \) |
| 97 | \( 1 + (-10.1 - 14.9i)T + (-3.48e3 + 8.74e3i)T^{2} \) |
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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
−11.86571361819147119915215237584, −11.09445548278316797972287838132, −10.19028419282223157374641953599, −9.781068139814709057861375175725, −8.731989483300788790197666028775, −6.40753565471141373596521738564, −5.83265051555058559977340465619, −4.07340094387984636301131857496, −2.49552745214860577588760308965, −1.13910309269257950238102710327,
1.38092541895413368701329772392, 4.83102680561515114299167472379, 5.52023187688808289101033698380, 6.57736004197280512099954200796, 6.98849556110401741932977356351, 8.767119259574002200946632547443, 9.505850179121224397792236944497, 10.33788655439766757867857410776, 11.88747616938079216255992007135, 13.19823089312912363653044309359
