| L(s) = 1 | + (−0.201 − 1.39i)2-s + (−2.17 + 4.75i)3-s + (−1.91 + 0.563i)4-s + (5.47 + 4.74i)5-s + (7.09 + 2.08i)6-s + (−0.874 − 1.36i)7-s + (1.17 + 2.57i)8-s + (−12.0 − 13.8i)9-s + (5.53 − 8.61i)10-s + (−4.71 − 0.678i)11-s + (1.48 − 10.3i)12-s + (16.1 + 10.3i)13-s + (−1.72 + 1.49i)14-s + (−34.4 + 15.7i)15-s + (3.36 − 2.16i)16-s + (8.15 − 27.7i)17-s + ⋯ |
| L(s) = 1 | + (−0.100 − 0.699i)2-s + (−0.724 + 1.58i)3-s + (−0.479 + 0.140i)4-s + (1.09 + 0.948i)5-s + (1.18 + 0.347i)6-s + (−0.124 − 0.194i)7-s + (0.146 + 0.321i)8-s + (−1.33 − 1.54i)9-s + (0.553 − 0.861i)10-s + (−0.428 − 0.0616i)11-s + (0.124 − 0.863i)12-s + (1.23 + 0.796i)13-s + (−0.123 + 0.107i)14-s + (−2.29 + 1.04i)15-s + (0.210 − 0.135i)16-s + (0.479 − 1.63i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.553 - 0.832i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.553 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.817200 + 0.438063i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.817200 + 0.438063i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.201 + 1.39i)T \) |
| 23 | \( 1 + (-10.1 + 20.6i)T \) |
| good | 3 | \( 1 + (2.17 - 4.75i)T + (-5.89 - 6.80i)T^{2} \) |
| 5 | \( 1 + (-5.47 - 4.74i)T + (3.55 + 24.7i)T^{2} \) |
| 7 | \( 1 + (0.874 + 1.36i)T + (-20.3 + 44.5i)T^{2} \) |
| 11 | \( 1 + (4.71 + 0.678i)T + (116. + 34.0i)T^{2} \) |
| 13 | \( 1 + (-16.1 - 10.3i)T + (70.2 + 153. i)T^{2} \) |
| 17 | \( 1 + (-8.15 + 27.7i)T + (-243. - 156. i)T^{2} \) |
| 19 | \( 1 + (-1.17 - 4.01i)T + (-303. + 195. i)T^{2} \) |
| 29 | \( 1 + (1.25 + 0.367i)T + (707. + 454. i)T^{2} \) |
| 31 | \( 1 + (-9.51 - 20.8i)T + (-629. + 726. i)T^{2} \) |
| 37 | \( 1 + (13.0 - 11.3i)T + (194. - 1.35e3i)T^{2} \) |
| 41 | \( 1 + (35.6 - 41.1i)T + (-239. - 1.66e3i)T^{2} \) |
| 43 | \( 1 + (-23.1 - 10.5i)T + (1.21e3 + 1.39e3i)T^{2} \) |
| 47 | \( 1 - 16.3T + 2.20e3T^{2} \) |
| 53 | \( 1 + (38.4 + 59.8i)T + (-1.16e3 + 2.55e3i)T^{2} \) |
| 59 | \( 1 + (67.7 + 43.5i)T + (1.44e3 + 3.16e3i)T^{2} \) |
| 61 | \( 1 + (19.1 - 8.73i)T + (2.43e3 - 2.81e3i)T^{2} \) |
| 67 | \( 1 + (29.8 - 4.29i)T + (4.30e3 - 1.26e3i)T^{2} \) |
| 71 | \( 1 + (10.0 + 70.0i)T + (-4.83e3 + 1.42e3i)T^{2} \) |
| 73 | \( 1 + (56.3 - 16.5i)T + (4.48e3 - 2.88e3i)T^{2} \) |
| 79 | \( 1 + (-2.09 + 3.25i)T + (-2.59e3 - 5.67e3i)T^{2} \) |
| 83 | \( 1 + (-49.8 + 43.2i)T + (980. - 6.81e3i)T^{2} \) |
| 89 | \( 1 + (-59.2 - 27.0i)T + (5.18e3 + 5.98e3i)T^{2} \) |
| 97 | \( 1 + (-38.1 - 33.0i)T + (1.33e3 + 9.31e3i)T^{2} \) |
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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
−15.93185286260133822137539857141, −14.49926495048979362933774621084, −13.60650951810259471468402934761, −11.66799249962948008886025514950, −10.74464763650923628462901380592, −10.05179359906494580942294574316, −9.105149093223908090661742182637, −6.37173143454229996415822153181, −4.94266093017177378708139990273, −3.20750974341348686214177720500,
1.36276074154779867721251276609, 5.59269270661318844236918182395, 6.07002843775429764102988767265, 7.71833925226223382618202629959, 8.834727388510410924248217754171, 10.62685554764795637971127118845, 12.40081431344983240192723509244, 13.12703874895401902472631711442, 13.73411609121918410527968816807, 15.56483333521747793395335102198
