L(s) = 1 | + (0.5 − 0.866i)3-s + (−1 − 1.73i)4-s + 3.46i·5-s + (−1.5 + 0.866i)7-s + (−0.499 − 0.866i)9-s + (−3 − 1.73i)11-s − 1.99·12-s + (3.5 − 0.866i)13-s + (2.99 + 1.73i)15-s + (−1.99 + 3.46i)16-s + (3 − 1.73i)19-s + (5.99 − 3.46i)20-s + 1.73i·21-s + (3 − 5.19i)23-s − 6.99·25-s + ⋯ |
L(s) = 1 | + (0.288 − 0.499i)3-s + (−0.5 − 0.866i)4-s + 1.54i·5-s + (−0.566 + 0.327i)7-s + (−0.166 − 0.288i)9-s + (−0.904 − 0.522i)11-s − 0.577·12-s + (0.970 − 0.240i)13-s + (0.774 + 0.447i)15-s + (−0.499 + 0.866i)16-s + (0.688 − 0.397i)19-s + (1.34 − 0.774i)20-s + 0.377i·21-s + (0.625 − 1.08i)23-s − 1.39·25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 + 0.265i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.964 + 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.729602 - 0.0984346i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.729602 - 0.0984346i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 13 | \( 1 + (-3.5 + 0.866i)T \) |
good | 2 | \( 1 + (1 + 1.73i)T^{2} \) |
| 5 | \( 1 - 3.46iT - 5T^{2} \) |
| 7 | \( 1 + (1.5 - 0.866i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (3 + 1.73i)T + (5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-3 + 1.73i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3 + 5.19i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (3 - 5.19i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 1.73iT - 31T^{2} \) |
| 37 | \( 1 + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (6 + 3.46i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.5 - 0.866i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 3.46iT - 47T^{2} \) |
| 53 | \( 1 - 12T + 53T^{2} \) |
| 59 | \( 1 + (3 - 1.73i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-7.5 - 4.33i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (9 - 5.19i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 - 1.73iT - 73T^{2} \) |
| 79 | \( 1 + 11T + 79T^{2} \) |
| 83 | \( 1 + 13.8iT - 83T^{2} \) |
| 89 | \( 1 + (-6 - 3.46i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (4.5 - 2.59i)T + (48.5 - 84.0i)T^{2} \) |
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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
−15.88808616152548985528097002510, −14.91846109816130859091273293097, −13.96419287562541034555919049480, −13.04619436231722837529409926035, −11.10551651873457236367808656979, −10.26712380940950273475966635755, −8.739124385087970338720574082298, −6.96894414488669299222772554564, −5.79122345317736035221054158129, −3.03451756138043372419512126667,
3.79028449147413468164063210595, 5.16881605044338984679923592521, 7.75582342999115831642782779487, 8.848690190067500122946250492219, 9.818472480147079412536725966363, 11.78493119263088006316411982323, 13.11314330705065684621775478908, 13.47313273791666928233950183414, 15.55977393109343258918665426919, 16.40034926286238632389372103235