L(s) = 1 | + (−2.44 − 4.58i)3-s + 2.31·5-s − 29.6i·7-s + (−15 + 22.4i)9-s − 22.8i·11-s + 8.66i·13-s + (−5.66 − 10.6i)15-s − 93.3i·17-s + 85.4·19-s + (−135. + 72.6i)21-s − 116.·23-s − 119.·25-s + (139. + 13.7i)27-s + 103.·29-s + 10.6i·31-s + ⋯ |
L(s) = 1 | + (−0.471 − 0.881i)3-s + 0.207·5-s − 1.60i·7-s + (−0.555 + 0.831i)9-s − 0.625i·11-s + 0.184i·13-s + (−0.0975 − 0.182i)15-s − 1.33i·17-s + 1.03·19-s + (−1.41 + 0.755i)21-s − 1.05·23-s − 0.957·25-s + (0.995 + 0.0979i)27-s + 0.662·29-s + 0.0614i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.956 - 0.290i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.956 - 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.8853803139\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8853803139\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (2.44 + 4.58i)T \) |
good | 5 | \( 1 - 2.31T + 125T^{2} \) |
| 7 | \( 1 + 29.6iT - 343T^{2} \) |
| 11 | \( 1 + 22.8iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 8.66iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 93.3iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 85.4T + 6.85e3T^{2} \) |
| 23 | \( 1 + 116.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 103.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 10.6iT - 2.97e4T^{2} \) |
| 37 | \( 1 - 380. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 257. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 359.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 293.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 679.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 45.8iT - 2.05e5T^{2} \) |
| 61 | \( 1 - 595. iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 206.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 996.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 593.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 910. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 332. iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 821. iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 420.T + 9.12e5T^{2} \) |
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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
−10.47199338592195463763979408888, −9.712843727158659097259485248913, −8.266125191507075324451643458084, −7.43590767448803334340380394894, −6.75650023170814952141581424845, −5.67480354333280835379596146327, −4.51085962044791509918175844042, −3.09047918561115116620530451857, −1.40947890281369833194396881850, −0.32578687765351167420783356335,
1.98384941106184680361136379435, 3.37516432675778798063781439050, 4.65757115680071180073991879191, 5.70942827648861987749794678777, 6.19968135295221766169740177497, 7.88062022343463798556631245511, 8.875302843814684706048608340830, 9.652706181143599101126496101139, 10.35272984731578702328951424974, 11.52664078709596289540860800920