L(s) = 1 | + (−1.72 − 0.143i)3-s + 2.77·5-s + (0.855 + 2.50i)7-s + (2.95 + 0.496i)9-s − 3.43·11-s + (−0.429 + 0.743i)13-s + (−4.78 − 0.398i)15-s + (−0.405 + 0.701i)17-s + (0.750 + 1.29i)19-s + (−1.11 − 4.44i)21-s + 7.64·23-s + 2.68·25-s + (−5.03 − 1.28i)27-s + (3.99 + 6.92i)29-s + (3.60 + 6.24i)31-s + ⋯ |
L(s) = 1 | + (−0.996 − 0.0829i)3-s + 1.23·5-s + (0.323 + 0.946i)7-s + (0.986 + 0.165i)9-s − 1.03·11-s + (−0.119 + 0.206i)13-s + (−1.23 − 0.102i)15-s + (−0.0982 + 0.170i)17-s + (0.172 + 0.298i)19-s + (−0.243 − 0.969i)21-s + 1.59·23-s + 0.536·25-s + (−0.969 − 0.246i)27-s + (0.742 + 1.28i)29-s + (0.647 + 1.12i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.698 - 0.715i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.698 - 0.715i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.16028 + 0.489174i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.16028 + 0.489174i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.72 + 0.143i)T \) |
| 7 | \( 1 + (-0.855 - 2.50i)T \) |
good | 5 | \( 1 - 2.77T + 5T^{2} \) |
| 11 | \( 1 + 3.43T + 11T^{2} \) |
| 13 | \( 1 + (0.429 - 0.743i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (0.405 - 0.701i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.750 - 1.29i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 - 7.64T + 23T^{2} \) |
| 29 | \( 1 + (-3.99 - 6.92i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-3.60 - 6.24i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.458 - 0.793i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.67 + 2.90i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.20 - 2.08i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-0.307 + 0.532i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-6.31 + 10.9i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (0.734 + 1.27i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (5.71 - 9.90i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (8.10 + 14.0i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 14.4T + 71T^{2} \) |
| 73 | \( 1 + (4.16 - 7.22i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.37 + 2.38i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (5.75 + 9.97i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (5.11 + 8.85i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.82 - 6.63i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.88135814670551639488316664504, −10.32054863773164736153418902688, −9.384329382652057471866997188169, −8.471910411354985573653120115910, −7.15089590783055931499147273772, −6.24407620298349468461838447894, −5.34307078875754758989336651829, −4.91057484814207918910826638827, −2.80507394242298386173203848515, −1.55119904023654387690287020757,
0.940268368206145131818672407331, 2.56130970897481284972711838510, 4.38056110744692985654657391384, 5.21736463833843051997639523336, 6.05695670203867152921420867772, 7.02026662196875239462132949689, 7.912019295039917191205933583865, 9.390910442366032378111731206956, 10.09443998367194065330103722588, 10.72107438471940664373632343016