| L(s) = 1 | + (3.00 − 0.530i)2-s + (−2.74 + 1.21i)3-s + (5.00 − 1.82i)4-s + (2.67 + 3.18i)5-s + (−7.60 + 5.11i)6-s + (4.42 + 7.65i)7-s + (3.50 − 2.02i)8-s + (6.03 − 6.67i)9-s + (9.73 + 8.17i)10-s − 5.92i·11-s + (−11.5 + 11.0i)12-s + (4.07 + 3.41i)13-s + (17.3 + 20.6i)14-s + (−11.2 − 5.48i)15-s + (−6.85 + 5.74i)16-s + (10.0 + 11.9i)17-s + ⋯ |
| L(s) = 1 | + (1.50 − 0.265i)2-s + (−0.914 + 0.405i)3-s + (1.25 − 0.455i)4-s + (0.535 + 0.637i)5-s + (−1.26 + 0.852i)6-s + (0.631 + 1.09i)7-s + (0.438 − 0.252i)8-s + (0.670 − 0.741i)9-s + (0.973 + 0.817i)10-s − 0.539i·11-s + (−0.958 + 0.923i)12-s + (0.313 + 0.262i)13-s + (1.23 + 1.47i)14-s + (−0.747 − 0.365i)15-s + (−0.428 + 0.359i)16-s + (0.590 + 0.703i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.828 - 0.559i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.828 - 0.559i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.61638 + 0.800287i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.61638 + 0.800287i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (2.74 - 1.21i)T \) |
| 19 | \( 1 + (-4.26 + 18.5i)T \) |
| good | 2 | \( 1 + (-3.00 + 0.530i)T + (3.75 - 1.36i)T^{2} \) |
| 5 | \( 1 + (-2.67 - 3.18i)T + (-4.34 + 24.6i)T^{2} \) |
| 7 | \( 1 + (-4.42 - 7.65i)T + (-24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + 5.92iT - 121T^{2} \) |
| 13 | \( 1 + (-4.07 - 3.41i)T + (29.3 + 166. i)T^{2} \) |
| 17 | \( 1 + (-10.0 - 11.9i)T + (-50.1 + 284. i)T^{2} \) |
| 23 | \( 1 + (5.66 + 15.5i)T + (-405. + 340. i)T^{2} \) |
| 29 | \( 1 + (13.1 + 36.2i)T + (-644. + 540. i)T^{2} \) |
| 31 | \( 1 + 29.0T + 961T^{2} \) |
| 37 | \( 1 - 23.7T + 1.36e3T^{2} \) |
| 41 | \( 1 + (32.1 - 5.67i)T + (1.57e3 - 574. i)T^{2} \) |
| 43 | \( 1 + (31.1 + 11.3i)T + (1.41e3 + 1.18e3i)T^{2} \) |
| 47 | \( 1 + (10.4 + 28.8i)T + (-1.69e3 + 1.41e3i)T^{2} \) |
| 53 | \( 1 + (-45.6 - 8.04i)T + (2.63e3 + 960. i)T^{2} \) |
| 59 | \( 1 + (-31.6 + 86.9i)T + (-2.66e3 - 2.23e3i)T^{2} \) |
| 61 | \( 1 + (-54.8 - 46.0i)T + (646. + 3.66e3i)T^{2} \) |
| 67 | \( 1 + (6.10 - 34.6i)T + (-4.21e3 - 1.53e3i)T^{2} \) |
| 71 | \( 1 + (-47.8 + 8.44i)T + (4.73e3 - 1.72e3i)T^{2} \) |
| 73 | \( 1 + (-112. - 40.9i)T + (4.08e3 + 3.42e3i)T^{2} \) |
| 79 | \( 1 + (111. - 93.3i)T + (1.08e3 - 6.14e3i)T^{2} \) |
| 83 | \( 1 + (-74.3 + 42.9i)T + (3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + (54.2 + 149. i)T + (-6.06e3 + 5.09e3i)T^{2} \) |
| 97 | \( 1 + (3.79 + 21.4i)T + (-8.84e3 + 3.21e3i)T^{2} \) |
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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
−12.56805534809827081087953526985, −11.61007916353160632890109036676, −11.17920586375552797467189713022, −10.02265908916255958522249536617, −8.621587284845065834976054614991, −6.63325287153720397606339620162, −5.83772989256544825855234624763, −5.12121926548341397069080160991, −3.80284296848695659982082437853, −2.30130691937076367147360339110,
1.44931224298017227032304730896, 3.82081241039868193474137093856, 5.04571406820826455596762453410, 5.55622476119967494773228507669, 6.89332294180296374170574331547, 7.72374642205816051567449601496, 9.657165097001026073974611557480, 10.83653903464223379015765357405, 11.80655894666063940995866121743, 12.65228617622953299850382642491
