| L(s) = 1 | + (−1.20 + 0.696i)2-s + (2.33 + 1.88i)3-s + (−1.03 + 1.78i)4-s + (−1.93 − 1.11i)5-s + (−4.12 − 0.653i)6-s + (4.41 + 7.63i)7-s − 8.43i·8-s + (1.87 + 8.80i)9-s + 3.11·10-s + (−0.805 + 0.464i)11-s + (−5.77 + 2.21i)12-s + (12.2 − 21.2i)13-s + (−10.6 − 6.13i)14-s + (−2.40 − 6.26i)15-s + (1.74 + 3.02i)16-s − 18.3i·17-s + ⋯ |
| L(s) = 1 | + (−0.602 + 0.348i)2-s + (0.777 + 0.629i)3-s + (−0.257 + 0.446i)4-s + (−0.387 − 0.223i)5-s + (−0.687 − 0.108i)6-s + (0.630 + 1.09i)7-s − 1.05i·8-s + (0.207 + 0.978i)9-s + 0.311·10-s + (−0.0732 + 0.0422i)11-s + (−0.481 + 0.184i)12-s + (0.941 − 1.63i)13-s + (−0.759 − 0.438i)14-s + (−0.160 − 0.417i)15-s + (0.109 + 0.189i)16-s − 1.07i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.139 - 0.990i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.139 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.722713 + 0.628270i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.722713 + 0.628270i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-2.33 - 1.88i)T \) |
| 5 | \( 1 + (1.93 + 1.11i)T \) |
| good | 2 | \( 1 + (1.20 - 0.696i)T + (2 - 3.46i)T^{2} \) |
| 7 | \( 1 + (-4.41 - 7.63i)T + (-24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (0.805 - 0.464i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (-12.2 + 21.2i)T + (-84.5 - 146. i)T^{2} \) |
| 17 | \( 1 + 18.3iT - 289T^{2} \) |
| 19 | \( 1 + 5.58T + 361T^{2} \) |
| 23 | \( 1 + (-20.6 - 11.9i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (23.7 - 13.7i)T + (420.5 - 728. i)T^{2} \) |
| 31 | \( 1 + (-4.66 + 8.08i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + 24.7T + 1.36e3T^{2} \) |
| 41 | \( 1 + (6.45 + 3.72i)T + (840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (17.7 + 30.7i)T + (-924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (0.298 - 0.172i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 - 81.8iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (65.9 + 38.0i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (29.6 + 51.3i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-40.9 + 70.9i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + 37.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 3.49T + 5.32e3T^{2} \) |
| 79 | \( 1 + (-62.0 - 107. i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-48.4 + 27.9i)T + (3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + 6.78iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (58.5 + 101. i)T + (-4.70e3 + 8.14e3i)T^{2} \) |
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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
−15.58006197432778027747079496110, −15.31261511852442513013119244595, −13.61996269057980693723490792220, −12.48475139847240222844561997916, −10.91225789386383821235213834448, −9.329404619943029915568050680404, −8.517172664464491131953656779968, −7.65829324598223460881858619451, −5.14774749823751520915131335268, −3.28849186857502195801927744384,
1.54096754891029680885690024283, 4.12618613471134001935278030738, 6.65849731611829411164142995831, 8.090766397792701151597580869428, 9.053103920113545748996303244640, 10.55063638541377352932737426806, 11.52084959118574622384973203556, 13.29596232400524264277318786139, 14.18996206139311359559180057437, 14.96928117416872928869062098057
