L(s) = 1 | + (2.85 − 0.920i)3-s + (−0.594 − 1.02i)7-s + (7.30 − 5.25i)9-s + (3.63 − 2.09i)11-s + (−7.15 + 12.3i)13-s + 18.7i·17-s + 33.8·19-s + (−2.64 − 2.39i)21-s + (11.6 + 6.74i)23-s + (16.0 − 21.7i)27-s + (30.6 − 17.6i)29-s + (4.97 − 8.62i)31-s + (8.44 − 9.33i)33-s + 19.3·37-s + (−9.02 + 41.9i)39-s + ⋯ |
L(s) = 1 | + (0.951 − 0.306i)3-s + (−0.0849 − 0.147i)7-s + (0.811 − 0.584i)9-s + (0.330 − 0.190i)11-s + (−0.550 + 0.953i)13-s + 1.10i·17-s + 1.78·19-s + (−0.125 − 0.113i)21-s + (0.508 + 0.293i)23-s + (0.593 − 0.805i)27-s + (1.05 − 0.609i)29-s + (0.160 − 0.278i)31-s + (0.255 − 0.282i)33-s + 0.521·37-s + (−0.231 + 1.07i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 + 0.246i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 900 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.969 + 0.246i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.878302148\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.878302148\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-2.85 + 0.920i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (0.594 + 1.02i)T + (-24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (-3.63 + 2.09i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (7.15 - 12.3i)T + (-84.5 - 146. i)T^{2} \) |
| 17 | \( 1 - 18.7iT - 289T^{2} \) |
| 19 | \( 1 - 33.8T + 361T^{2} \) |
| 23 | \( 1 + (-11.6 - 6.74i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (-30.6 + 17.6i)T + (420.5 - 728. i)T^{2} \) |
| 31 | \( 1 + (-4.97 + 8.62i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 - 19.3T + 1.36e3T^{2} \) |
| 41 | \( 1 + (55.9 + 32.3i)T + (840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (20.7 + 35.9i)T + (-924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-58.2 + 33.6i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 - 30.0iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (-3.66 - 2.11i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-43.8 - 75.9i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (17.9 - 31.0i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + 24.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 11.9T + 5.32e3T^{2} \) |
| 79 | \( 1 + (-19.0 - 33.0i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (20.8 - 12.0i)T + (3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + 44.4iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-31.9 - 55.3i)T + (-4.70e3 + 8.14e3i)T^{2} \) |
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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
−9.757548587439649465065106492962, −8.995671482764581155684010288836, −8.277466615888309603724877411725, −7.29074783847524542802336044852, −6.75164718217858146571142570622, −5.53184287941545043324383163437, −4.26401356152411208763868510790, −3.44718736243029650077319285089, −2.30036573524464547069038900167, −1.11517477485590296002928295154,
1.11265401217483346983450565177, 2.73641931983528551985856186637, 3.24580415646980677519673895731, 4.66219005321388122572685381385, 5.29705151755803214109905743598, 6.76061308292081529404362388355, 7.52945359160108280544577643187, 8.255504967023463378308350253791, 9.273612396533837125849284472325, 9.737510258062827066100983743860