L(s) = 1 | + (−1 + 1.73i)3-s + (−0.5 − 0.866i)5-s + (−0.499 − 0.866i)9-s + (0.5 − 0.866i)11-s + 3·13-s + 1.99·15-s + (−1 + 1.73i)17-s + (−2.5 − 4.33i)19-s + (−3.5 − 6.06i)23-s + (−0.499 + 0.866i)25-s − 4.00·27-s − 6·29-s + (2 − 3.46i)31-s + (0.999 + 1.73i)33-s + (2.5 + 4.33i)37-s + ⋯ |
L(s) = 1 | + (−0.577 + 0.999i)3-s + (−0.223 − 0.387i)5-s + (−0.166 − 0.288i)9-s + (0.150 − 0.261i)11-s + 0.832·13-s + 0.516·15-s + (−0.242 + 0.420i)17-s + (−0.573 − 0.993i)19-s + (−0.729 − 1.26i)23-s + (−0.0999 + 0.173i)25-s − 0.769·27-s − 1.11·29-s + (0.359 − 0.622i)31-s + (0.174 + 0.301i)33-s + (0.410 + 0.711i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9066367587\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9066367587\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (1 - 1.73i)T + (-1.5 - 2.59i)T^{2} \) |
| 11 | \( 1 + (-0.5 + 0.866i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 3T + 13T^{2} \) |
| 17 | \( 1 + (1 - 1.73i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2.5 + 4.33i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3.5 + 6.06i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + (-2 + 3.46i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.5 - 4.33i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 5T + 41T^{2} \) |
| 43 | \( 1 - 6T + 43T^{2} \) |
| 47 | \( 1 + (4.5 + 7.79i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (5.5 - 9.52i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-4 + 6.92i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (6 + 10.3i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-2 + 3.46i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 4T + 71T^{2} \) |
| 73 | \( 1 + (-6 + 10.3i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (7 + 12.1i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 + (-3 - 5.19i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 6T + 97T^{2} \) |
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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
−9.134081321630978594977132083773, −8.428784275500972841118872782813, −7.63102494361054254692186417646, −6.38085062405541775631951690032, −5.90861127441862910397381835284, −4.77631265998901889030915020096, −4.33118417454365400517022232495, −3.45768349910297191515617908531, −2.04064952679526463951370381090, −0.38991862446237965611773256504,
1.17397570701143918203395662333, 2.13006107416386283590282070532, 3.50311324456330301325603241294, 4.28081560281998341919468480497, 5.73002376479802169829029793518, 6.04368449582268823240536042515, 7.01725939711515055639824937836, 7.53689981091878564986593317378, 8.306200387651408987341462356947, 9.321651539676489981663414754950