L(s) = 1 | + (0.923 − 0.382i)3-s + (0.184 − 0.444i)5-s + (0.134 + 0.134i)7-s + (0.707 − 0.707i)9-s + (4.83 + 2.00i)11-s + (−0.237 − 0.573i)13-s − 0.480i·15-s − 5.70i·17-s + (0.459 + 1.10i)19-s + (0.175 + 0.0728i)21-s + (4.07 − 4.07i)23-s + (3.37 + 3.37i)25-s + (0.382 − 0.923i)27-s + (−2.33 + 0.966i)29-s − 10.2·31-s + ⋯ |
L(s) = 1 | + (0.533 − 0.220i)3-s + (0.0823 − 0.198i)5-s + (0.0508 + 0.0508i)7-s + (0.235 − 0.235i)9-s + (1.45 + 0.603i)11-s + (−0.0658 − 0.159i)13-s − 0.124i·15-s − 1.38i·17-s + (0.105 + 0.254i)19-s + (0.0383 + 0.0158i)21-s + (0.850 − 0.850i)23-s + (0.674 + 0.674i)25-s + (0.0736 − 0.177i)27-s + (−0.433 + 0.179i)29-s − 1.83·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.936 + 0.351i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.936 + 0.351i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.70773 - 0.310034i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.70773 - 0.310034i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.923 + 0.382i)T \) |
good | 5 | \( 1 + (-0.184 + 0.444i)T + (-3.53 - 3.53i)T^{2} \) |
| 7 | \( 1 + (-0.134 - 0.134i)T + 7iT^{2} \) |
| 11 | \( 1 + (-4.83 - 2.00i)T + (7.77 + 7.77i)T^{2} \) |
| 13 | \( 1 + (0.237 + 0.573i)T + (-9.19 + 9.19i)T^{2} \) |
| 17 | \( 1 + 5.70iT - 17T^{2} \) |
| 19 | \( 1 + (-0.459 - 1.10i)T + (-13.4 + 13.4i)T^{2} \) |
| 23 | \( 1 + (-4.07 + 4.07i)T - 23iT^{2} \) |
| 29 | \( 1 + (2.33 - 0.966i)T + (20.5 - 20.5i)T^{2} \) |
| 31 | \( 1 + 10.2T + 31T^{2} \) |
| 37 | \( 1 + (3.05 - 7.37i)T + (-26.1 - 26.1i)T^{2} \) |
| 41 | \( 1 + (0.877 - 0.877i)T - 41iT^{2} \) |
| 43 | \( 1 + (-2.15 - 0.893i)T + (30.4 + 30.4i)T^{2} \) |
| 47 | \( 1 + 4.94iT - 47T^{2} \) |
| 53 | \( 1 + (-9.74 - 4.03i)T + (37.4 + 37.4i)T^{2} \) |
| 59 | \( 1 + (4.73 - 11.4i)T + (-41.7 - 41.7i)T^{2} \) |
| 61 | \( 1 + (9.46 - 3.92i)T + (43.1 - 43.1i)T^{2} \) |
| 67 | \( 1 + (4.79 - 1.98i)T + (47.3 - 47.3i)T^{2} \) |
| 71 | \( 1 + (4.32 + 4.32i)T + 71iT^{2} \) |
| 73 | \( 1 + (6.12 - 6.12i)T - 73iT^{2} \) |
| 79 | \( 1 - 11.9iT - 79T^{2} \) |
| 83 | \( 1 + (1.59 + 3.86i)T + (-58.6 + 58.6i)T^{2} \) |
| 89 | \( 1 + (8.43 + 8.43i)T + 89iT^{2} \) |
| 97 | \( 1 + 9.21T + 97T^{2} \) |
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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
−11.44933237081876988271358002998, −10.25150357411831316131676486624, −9.158096641517046465056392073530, −8.849839145135126936273427361307, −7.34106396002709057588087797114, −6.85104997781084776124917784840, −5.39327884477050989736228801741, −4.24387137602891532316532333764, −2.98217856182731216419603402603, −1.44337242133926301487863037248,
1.67015478280331614974741485914, 3.33874802905927312881759088766, 4.17087040569845628512505813647, 5.65626982860170755011043010355, 6.68719816352671755460924147435, 7.68722300268713735248353331920, 8.924423793830576181541277026088, 9.249922280726533180464202651758, 10.59765892668586256168643514117, 11.19729221655615839318946961150