L(s) = 1 | + (0.707 + 0.707i)3-s + (−1.32 + 1.80i)5-s + (1.86 − 1.86i)7-s + 1.00i·9-s − 0.728i·11-s + (3.12 − 3.12i)13-s + (−2.20 + 0.342i)15-s + (1.12 + 1.12i)17-s + 3.73·19-s + 2.64·21-s + (5.83 + 5.83i)23-s + (−1.51 − 4.76i)25-s + (−0.707 + 0.707i)27-s + 2.64i·29-s + 6.01i·31-s + ⋯ |
L(s) = 1 | + (0.408 + 0.408i)3-s + (−0.590 + 0.807i)5-s + (0.705 − 0.705i)7-s + 0.333i·9-s − 0.219i·11-s + (0.866 − 0.866i)13-s + (−0.570 + 0.0885i)15-s + (0.272 + 0.272i)17-s + 0.856·19-s + 0.576·21-s + (1.21 + 1.21i)23-s + (−0.303 − 0.952i)25-s + (−0.136 + 0.136i)27-s + 0.490i·29-s + 1.07i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.759 - 0.649i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.759 - 0.649i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.76891 + 0.653239i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.76891 + 0.653239i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.707 - 0.707i)T \) |
| 5 | \( 1 + (1.32 - 1.80i)T \) |
good | 7 | \( 1 + (-1.86 + 1.86i)T - 7iT^{2} \) |
| 11 | \( 1 + 0.728iT - 11T^{2} \) |
| 13 | \( 1 + (-3.12 + 3.12i)T - 13iT^{2} \) |
| 17 | \( 1 + (-1.12 - 1.12i)T + 17iT^{2} \) |
| 19 | \( 1 - 3.73T + 19T^{2} \) |
| 23 | \( 1 + (-5.83 - 5.83i)T + 23iT^{2} \) |
| 29 | \( 1 - 2.64iT - 29T^{2} \) |
| 31 | \( 1 - 6.01iT - 31T^{2} \) |
| 37 | \( 1 + (3.12 + 3.12i)T + 37iT^{2} \) |
| 41 | \( 1 + 4.24T + 41T^{2} \) |
| 43 | \( 1 + (-5.10 - 5.10i)T + 43iT^{2} \) |
| 47 | \( 1 + (2.09 - 2.09i)T - 47iT^{2} \) |
| 53 | \( 1 + (0.484 - 0.484i)T - 53iT^{2} \) |
| 59 | \( 1 - 4.92T + 59T^{2} \) |
| 61 | \( 1 + 2.31T + 61T^{2} \) |
| 67 | \( 1 + (5.10 - 5.10i)T - 67iT^{2} \) |
| 71 | \( 1 + 13.1iT - 71T^{2} \) |
| 73 | \( 1 + (-3.96 + 3.96i)T - 73iT^{2} \) |
| 79 | \( 1 - 7.11T + 79T^{2} \) |
| 83 | \( 1 + (-3.55 - 3.55i)T + 83iT^{2} \) |
| 89 | \( 1 + 1.03iT - 89T^{2} \) |
| 97 | \( 1 + (12.5 + 12.5i)T + 97iT^{2} \) |
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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
−10.34535893077547394826138931154, −9.276744018515309910974220348583, −8.316620772046030019705400697634, −7.65627021898451910354257669435, −6.99206176231691989573696011203, −5.71723024155842567499205775521, −4.74872201429807708008071960061, −3.55556808965196796818498069446, −3.13022714980921845150388832238, −1.28668591331309933325461287384,
1.06796706465958162340596255383, 2.27320277860362826138128046133, 3.62795440887873634509493742734, 4.65333373936817647315363280196, 5.46966643193608322486361991562, 6.64741899382456178285736345779, 7.56340589762472689471903930303, 8.414403120470782406223396068756, 8.855563579581167267142844802779, 9.650774088222718236946518498825