L(s) = 1 | + (−1.82 − 0.813i)2-s + (−0.851 − 0.491i)3-s + (2.67 + 2.97i)4-s + (1.71 − 2.97i)5-s + (1.15 + 1.59i)6-s − 2.43i·7-s + (−2.47 − 7.60i)8-s + (−4.01 − 6.95i)9-s + (−5.55 + 4.03i)10-s − 7.76i·11-s + (−0.819 − 3.84i)12-s + (−4.63 − 8.02i)13-s + (−1.98 + 4.45i)14-s + (−2.92 + 1.68i)15-s + (−1.65 + 15.9i)16-s + (8.87 − 15.3i)17-s + ⋯ |
L(s) = 1 | + (−0.913 − 0.406i)2-s + (−0.283 − 0.163i)3-s + (0.669 + 0.742i)4-s + (0.343 − 0.594i)5-s + (0.192 + 0.265i)6-s − 0.348i·7-s + (−0.309 − 0.950i)8-s + (−0.446 − 0.772i)9-s + (−0.555 + 0.403i)10-s − 0.705i·11-s + (−0.0682 − 0.320i)12-s + (−0.356 − 0.617i)13-s + (−0.141 + 0.318i)14-s + (−0.194 + 0.112i)15-s + (−0.103 + 0.994i)16-s + (0.521 − 0.903i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.252 + 0.967i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.252 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.443066 - 0.573798i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.443066 - 0.573798i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.82 + 0.813i)T \) |
| 19 | \( 1 + (-6.84 - 17.7i)T \) |
good | 3 | \( 1 + (0.851 + 0.491i)T + (4.5 + 7.79i)T^{2} \) |
| 5 | \( 1 + (-1.71 + 2.97i)T + (-12.5 - 21.6i)T^{2} \) |
| 7 | \( 1 + 2.43iT - 49T^{2} \) |
| 11 | \( 1 + 7.76iT - 121T^{2} \) |
| 13 | \( 1 + (4.63 + 8.02i)T + (-84.5 + 146. i)T^{2} \) |
| 17 | \( 1 + (-8.87 + 15.3i)T + (-144.5 - 250. i)T^{2} \) |
| 23 | \( 1 + (-10.0 + 5.80i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (-2.64 - 4.57i)T + (-420.5 + 728. i)T^{2} \) |
| 31 | \( 1 - 10.2iT - 961T^{2} \) |
| 37 | \( 1 + 7.71T + 1.36e3T^{2} \) |
| 41 | \( 1 + (25.2 - 43.6i)T + (-840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (43.2 + 24.9i)T + (924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-74.9 + 43.2i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-17.3 - 30.0i)T + (-1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (63.9 + 36.9i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-40.4 - 70.0i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (14.8 - 8.60i)T + (2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + (-47.9 - 27.6i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-27.9 + 48.4i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-95.2 - 54.9i)T + (3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 - 31.3iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (5.29 + 9.17i)T + (-3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-65.4 + 113. i)T + (-4.70e3 - 8.14e3i)T^{2} \) |
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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
−13.75374902801219529481851771376, −12.51138063869937887739048971803, −11.74789255175919654941053993035, −10.51627593792092198429641920302, −9.423596435943806649051752047843, −8.413097195590370032030712098397, −7.06938297098523276287044450720, −5.57108987740095349188803499864, −3.26257676113740747121245133911, −0.886182368673908640143426385684,
2.31397859177572128822767624451, 5.14402459312669807650555285674, 6.44817487671988377595283741410, 7.62081515492945318404607805929, 8.968176478705099918253162999834, 10.11042527186006791150829444985, 10.95733439325103938463504433870, 12.07860716238944429153833376475, 13.81910869723554128765050120654, 14.80022411023744713529388678223