L(s) = 1 | + (0.809 + 0.587i)2-s + (−1.57 + 0.714i)3-s + (0.309 + 0.951i)4-s + (−1.72 − 2.37i)5-s + (−1.69 − 0.349i)6-s + (0.951 − 0.309i)7-s + (−0.309 + 0.951i)8-s + (1.97 − 2.25i)9-s − 2.93i·10-s + (−3.22 + 0.794i)11-s + (−1.16 − 1.27i)12-s + (3.69 − 5.08i)13-s + (0.951 + 0.309i)14-s + (4.41 + 2.51i)15-s + (−0.809 + 0.587i)16-s + (4.68 − 3.40i)17-s + ⋯ |
L(s) = 1 | + (0.572 + 0.415i)2-s + (−0.911 + 0.412i)3-s + (0.154 + 0.475i)4-s + (−0.770 − 1.06i)5-s + (−0.692 − 0.142i)6-s + (0.359 − 0.116i)7-s + (−0.109 + 0.336i)8-s + (0.659 − 0.751i)9-s − 0.927i·10-s + (−0.970 + 0.239i)11-s + (−0.336 − 0.369i)12-s + (1.02 − 1.40i)13-s + (0.254 + 0.0825i)14-s + (1.13 + 0.648i)15-s + (−0.202 + 0.146i)16-s + (1.13 − 0.824i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.790 + 0.612i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.790 + 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.10147 - 0.376775i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.10147 - 0.376775i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.809 - 0.587i)T \) |
| 3 | \( 1 + (1.57 - 0.714i)T \) |
| 7 | \( 1 + (-0.951 + 0.309i)T \) |
| 11 | \( 1 + (3.22 - 0.794i)T \) |
good | 5 | \( 1 + (1.72 + 2.37i)T + (-1.54 + 4.75i)T^{2} \) |
| 13 | \( 1 + (-3.69 + 5.08i)T + (-4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-4.68 + 3.40i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (-0.848 - 0.275i)T + (15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + 5.08iT - 23T^{2} \) |
| 29 | \( 1 + (-1.40 - 4.31i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (5.16 + 3.75i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-0.894 - 2.75i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (-3.18 + 9.80i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + 2.94iT - 43T^{2} \) |
| 47 | \( 1 + (-3.42 - 1.11i)T + (38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (4.17 - 5.74i)T + (-16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (10.4 - 3.40i)T + (47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (2.22 + 3.06i)T + (-18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 - 4.70T + 67T^{2} \) |
| 71 | \( 1 + (-0.962 - 1.32i)T + (-21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (8.25 - 2.68i)T + (59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (-7.37 + 10.1i)T + (-24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (2.82 - 2.05i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + 1.62iT - 89T^{2} \) |
| 97 | \( 1 + (15.5 + 11.3i)T + (29.9 + 92.2i)T^{2} \) |
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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
−10.97702307944568543309109063545, −10.37922874452569583472329811817, −9.015499963493239251370503410965, −8.017536590492183173106141753440, −7.37041333576585955667638898549, −5.83857922660080296223163188308, −5.23401303085124600652218397938, −4.45350399447655111277756725275, −3.32164713369565124374040707031, −0.72170536710544463676826557263,
1.63439343673772837653250007706, 3.23609936737501552309746997274, 4.29203602390421840229755534347, 5.54381949187350132742553900083, 6.33953831507822963593312555779, 7.33789101959907897658607109131, 8.103533990006448093339025425191, 9.723139973063441457097653615914, 10.83147000514636328731143703582, 11.15150484053256439874862469929