| L(s) = 1 | + (−0.707 + 1.22i)2-s + (−1.5 + 0.866i)3-s + (−0.999 − 1.73i)4-s − 2.44i·6-s + (−1.74 − 6.77i)7-s + 2.82·8-s + (1.5 − 2.59i)9-s + (−3 − 5.19i)11-s + (2.99 + 1.73i)12-s − 21.3i·13-s + (9.53 + 2.65i)14-s + (−2.00 + 3.46i)16-s + (7.75 − 4.47i)17-s + (2.12 + 3.67i)18-s + (−6.25 − 3.61i)19-s + ⋯ |
| L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.5 + 0.288i)3-s + (−0.249 − 0.433i)4-s − 0.408i·6-s + (−0.248 − 0.968i)7-s + 0.353·8-s + (0.166 − 0.288i)9-s + (−0.272 − 0.472i)11-s + (0.249 + 0.144i)12-s − 1.64i·13-s + (0.681 + 0.189i)14-s + (−0.125 + 0.216i)16-s + (0.456 − 0.263i)17-s + (0.117 + 0.204i)18-s + (−0.329 − 0.190i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 + 0.124i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.992 + 0.124i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.06072108278\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.06072108278\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.707 - 1.22i)T \) |
| 3 | \( 1 + (1.5 - 0.866i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (1.74 + 6.77i)T \) |
| good | 11 | \( 1 + (3 + 5.19i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + 21.3iT - 169T^{2} \) |
| 17 | \( 1 + (-7.75 + 4.47i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (6.25 + 3.61i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (18.7 - 32.4i)T + (-264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + 33.9T + 841T^{2} \) |
| 31 | \( 1 + (-38.2 + 22.0i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (13.9 - 24.2i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 - 54.8iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 1.48T + 1.84e3T^{2} \) |
| 47 | \( 1 + (-37.2 - 21.5i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (42.7 + 74.0i)T + (-1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (35.6 - 20.6i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (1.02 + 0.594i)T + (1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-2.19 - 3.80i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 137.T + 5.04e3T^{2} \) |
| 73 | \( 1 + (68.3 - 39.4i)T + (2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (49.1 - 85.1i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 - 110. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (18 + 10.3i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + 10.9iT - 9.40e3T^{2} \) |
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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
−9.652902773708568523665002081765, −8.148192049815874935301826212591, −7.80962623929758720815744778341, −6.79400476556162287912653337171, −5.85964530137194389945831537455, −5.24883035471263320362371983261, −4.08717860389135144045866247312, −3.09093433431019146863248351869, −1.09990514676108090002932623513, −0.02569986684135897953863664692,
1.73054114101247446634974450413, 2.46240205158143096245942205697, 3.91879207265715717443316867654, 4.83695083046739271996152300861, 5.95711397906995359105469548077, 6.71194662442022795916198891626, 7.69721501972376764873984091343, 8.701247657822608858850312842029, 9.264352220366203693756186260963, 10.20640511861099407924904926173
