L(s) = 1 | + (2 + 2.23i)3-s − 4i·5-s − 8.94·7-s + (−1.00 + 8.94i)9-s + 4.47i·11-s + 17.8·13-s + (8.94 − 8i)15-s + 17.8i·17-s − 20·19-s + (−17.8 − 20.0i)21-s − 16i·23-s + 9·25-s + (−22.0 + 15.6i)27-s + 52i·29-s − 26.8·31-s + ⋯ |
L(s) = 1 | + (0.666 + 0.745i)3-s − 0.800i·5-s − 1.27·7-s + (−0.111 + 0.993i)9-s + 0.406i·11-s + 1.37·13-s + (0.596 − 0.533i)15-s + 1.05i·17-s − 1.05·19-s + (−0.851 − 0.952i)21-s − 0.695i·23-s + 0.359·25-s + (−0.814 + 0.579i)27-s + 1.79i·29-s − 0.865·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.666 - 0.745i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.666 - 0.745i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.277492220\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.277492220\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-2 - 2.23i)T \) |
good | 5 | \( 1 + 4iT - 25T^{2} \) |
| 7 | \( 1 + 8.94T + 49T^{2} \) |
| 11 | \( 1 - 4.47iT - 121T^{2} \) |
| 13 | \( 1 - 17.8T + 169T^{2} \) |
| 17 | \( 1 - 17.8iT - 289T^{2} \) |
| 19 | \( 1 + 20T + 361T^{2} \) |
| 23 | \( 1 + 16iT - 529T^{2} \) |
| 29 | \( 1 - 52iT - 841T^{2} \) |
| 31 | \( 1 + 26.8T + 961T^{2} \) |
| 37 | \( 1 + 53.6T + 1.36e3T^{2} \) |
| 41 | \( 1 - 35.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 36T + 1.84e3T^{2} \) |
| 47 | \( 1 - 64iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 20iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 102. iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 17.8T + 3.72e3T^{2} \) |
| 67 | \( 1 + 44T + 4.48e3T^{2} \) |
| 71 | \( 1 + 80iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 50T + 5.32e3T^{2} \) |
| 79 | \( 1 - 80.4T + 6.24e3T^{2} \) |
| 83 | \( 1 - 102. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 160. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 50T + 9.40e3T^{2} \) |
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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
−10.51096789722698517631362910994, −9.397794551090893375740933539756, −8.779719967446302176218455447036, −8.299385815396771177564719336001, −6.88790177069895196943270257360, −6.01333714643961216195995043768, −4.87063134985177048097897712322, −3.90106535502129856597394102978, −3.15558779846777955086525971210, −1.62023312677747646096731307270,
0.38781954531104014354274751066, 2.10765152338605495486233640089, 3.23740409400839883690108703013, 3.74512100046242157105018741524, 5.69852322831963765490257175166, 6.58707751752186665155664976110, 6.95776691417840521613779656667, 8.123691181287954817794636783696, 8.923208523687528338481276832373, 9.686360342362426487152943754255