| L(s) = 1 | + (0.860 + 1.12i)2-s + (1.59 − 0.669i)3-s + (−0.517 + 1.93i)4-s + (3.00 − 1.09i)5-s + (2.12 + 1.21i)6-s + (−4.58 + 0.808i)7-s + (−2.61 + 1.08i)8-s + (2.10 − 2.13i)9-s + (3.81 + 2.43i)10-s + (−0.303 + 0.833i)11-s + (0.466 + 3.43i)12-s + (1.22 − 1.45i)13-s + (−4.85 − 4.44i)14-s + (4.07 − 3.76i)15-s + (−3.46 − 2.00i)16-s + (−4.03 − 2.32i)17-s + ⋯ |
| L(s) = 1 | + (0.608 + 0.793i)2-s + (0.922 − 0.386i)3-s + (−0.258 + 0.965i)4-s + (1.34 − 0.489i)5-s + (0.868 + 0.496i)6-s + (−1.73 + 0.305i)7-s + (−0.923 + 0.382i)8-s + (0.700 − 0.713i)9-s + (1.20 + 0.769i)10-s + (−0.0914 + 0.251i)11-s + (0.134 + 0.990i)12-s + (0.338 − 0.403i)13-s + (−1.29 − 1.18i)14-s + (1.05 − 0.972i)15-s + (−0.865 − 0.500i)16-s + (−0.978 − 0.564i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.718 - 0.695i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.718 - 0.695i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.94266 + 0.786130i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.94266 + 0.786130i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.860 - 1.12i)T \) |
| 3 | \( 1 + (-1.59 + 0.669i)T \) |
| good | 5 | \( 1 + (-3.00 + 1.09i)T + (3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (4.58 - 0.808i)T + (6.57 - 2.39i)T^{2} \) |
| 11 | \( 1 + (0.303 - 0.833i)T + (-8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (-1.22 + 1.45i)T + (-2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (4.03 + 2.32i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.171 + 0.296i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.00 - 5.68i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (4.16 - 3.49i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-7.80 - 1.37i)T + (29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (2.31 + 1.33i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.0346 + 0.0413i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (2.85 + 1.04i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (1.90 + 10.7i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 - 1.27T + 53T^{2} \) |
| 59 | \( 1 + (-2.43 - 6.68i)T + (-45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-8.19 + 1.44i)T + (57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (6.31 + 5.29i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (0.186 - 0.323i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-6.29 - 10.9i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (2.54 + 3.03i)T + (-13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-8.90 - 10.6i)T + (-14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (6.35 - 3.66i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (0.833 + 0.303i)T + (74.3 + 62.3i)T^{2} \) |
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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
−13.07496110283674069404977693182, −12.07251889984841094923927413590, −9.957324805048032362580976156204, −9.306271891809551095358210460960, −8.581829188054038853489100922099, −7.07767258233656024621715621033, −6.36549241535767567746972746617, −5.34616452591745069160582979217, −3.61008822627537983102100157132, −2.45703510623619484586395576517,
2.21138235873014524389658998104, 3.14985384986433858577511939588, 4.32381794982433439712269841196, 6.07664283168937633897278987357, 6.63777244260437645545732789797, 8.738815764557886445576591034861, 9.679504552079187960618317067397, 10.08597101414650221518497895079, 10.93812798211978437659250721562, 12.61407967502491980082857015599
