L(s) = 1 | + (0.309 + 0.224i)2-s + (−0.309 + 0.951i)3-s + (−0.572 − 1.76i)4-s + (−1.30 + 0.951i)5-s + (−0.309 + 0.224i)6-s + (−0.309 − 0.951i)7-s + (0.454 − 1.40i)8-s + (−0.809 − 0.587i)9-s − 0.618·10-s + (−2.19 + 2.48i)11-s + 1.85·12-s + (3.42 + 2.48i)13-s + (0.118 − 0.363i)14-s + (−0.499 − 1.53i)15-s + (−2.54 + 1.84i)16-s + (6.35 − 4.61i)17-s + ⋯ |
L(s) = 1 | + (0.218 + 0.158i)2-s + (−0.178 + 0.549i)3-s + (−0.286 − 0.881i)4-s + (−0.585 + 0.425i)5-s + (−0.126 + 0.0916i)6-s + (−0.116 − 0.359i)7-s + (0.160 − 0.495i)8-s + (−0.269 − 0.195i)9-s − 0.195·10-s + (−0.660 + 0.750i)11-s + 0.535·12-s + (0.950 + 0.690i)13-s + (0.0315 − 0.0970i)14-s + (−0.129 − 0.397i)15-s + (−0.636 + 0.462i)16-s + (1.54 − 1.11i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.242i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 - 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.685332 + 0.0842454i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.685332 + 0.0842454i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.309 - 0.951i)T \) |
| 11 | \( 1 + (2.19 - 2.48i)T \) |
good | 2 | \( 1 + (-0.309 - 0.224i)T + (0.618 + 1.90i)T^{2} \) |
| 5 | \( 1 + (1.30 - 0.951i)T + (1.54 - 4.75i)T^{2} \) |
| 7 | \( 1 + (0.309 + 0.951i)T + (-5.66 + 4.11i)T^{2} \) |
| 13 | \( 1 + (-3.42 - 2.48i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (-6.35 + 4.61i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (0.263 - 0.812i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + 4.23T + 23T^{2} \) |
| 29 | \( 1 + (1.85 + 5.70i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (4.11 + 2.99i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (0.545 + 1.67i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (1.30 - 4.02i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 - 6.70T + 43T^{2} \) |
| 47 | \( 1 + (-0.336 + 1.03i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-2.11 - 1.53i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (-2.97 - 9.14i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (6.92 - 5.03i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + 4.85T + 67T^{2} \) |
| 71 | \( 1 + (-4.30 + 3.13i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-2.38 - 7.33i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (8.89 + 6.46i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (-6.04 + 4.39i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + 3.76T + 89T^{2} \) |
| 97 | \( 1 + (0.927 + 0.673i)T + (29.9 + 92.2i)T^{2} \) |
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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
−16.48660660504225864507154828962, −15.57961920766766971037884281083, −14.59678019166787755615263599236, −13.52849772848038552864046426470, −11.73748930687682857742956772434, −10.48627858732892218675964487356, −9.496414475514369998389671803402, −7.44629319321667452396416432202, −5.70602665208127893703918281374, −4.07106803246905813455387311145,
3.50615854187679487427044393002, 5.65105796164047322365801015824, 7.82752521761884850187564116000, 8.553772017013687259261755834245, 10.81741248565105706753367219342, 12.22878587209579477609451956978, 12.80408060483079045697412295504, 14.05339806885900157386526069329, 15.82376536896543676893985401643, 16.69140000180274586172293298347