| L(s) = 1 | + (−1.18 − 0.775i)2-s + (−0.200 + 1.72i)3-s + (0.796 + 1.83i)4-s + (−0.263 − 2.00i)5-s + (1.57 − 1.87i)6-s + (−0.327 − 1.22i)7-s + (0.482 − 2.78i)8-s + (−2.91 − 0.690i)9-s + (−1.24 + 2.57i)10-s + (4.33 − 3.32i)11-s + (−3.31 + 1.00i)12-s + (−1.58 + 2.05i)13-s + (−0.560 + 1.69i)14-s + (3.49 − 0.0516i)15-s + (−2.73 + 2.92i)16-s + 4.55·17-s + ⋯ |
| L(s) = 1 | + (−0.836 − 0.548i)2-s + (−0.115 + 0.993i)3-s + (0.398 + 0.917i)4-s + (−0.117 − 0.895i)5-s + (0.641 − 0.766i)6-s + (−0.123 − 0.461i)7-s + (0.170 − 0.985i)8-s + (−0.973 − 0.230i)9-s + (−0.392 + 0.813i)10-s + (1.30 − 1.00i)11-s + (−0.957 + 0.289i)12-s + (−0.438 + 0.571i)13-s + (−0.149 + 0.453i)14-s + (0.903 − 0.0133i)15-s + (−0.683 + 0.730i)16-s + 1.10·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.726 + 0.687i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.726 + 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.773174 - 0.307784i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.773174 - 0.307784i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.18 + 0.775i)T \) |
| 3 | \( 1 + (0.200 - 1.72i)T \) |
| good | 5 | \( 1 + (0.263 + 2.00i)T + (-4.82 + 1.29i)T^{2} \) |
| 7 | \( 1 + (0.327 + 1.22i)T + (-6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (-4.33 + 3.32i)T + (2.84 - 10.6i)T^{2} \) |
| 13 | \( 1 + (1.58 - 2.05i)T + (-3.36 - 12.5i)T^{2} \) |
| 17 | \( 1 - 4.55T + 17T^{2} \) |
| 19 | \( 1 + (-5.75 + 2.38i)T + (13.4 - 13.4i)T^{2} \) |
| 23 | \( 1 + (-2.66 - 0.713i)T + (19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (9.89 + 1.30i)T + (28.0 + 7.50i)T^{2} \) |
| 31 | \( 1 + (-3.42 + 1.97i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (2.63 - 6.35i)T + (-26.1 - 26.1i)T^{2} \) |
| 41 | \( 1 + (-2.39 + 8.95i)T + (-35.5 - 20.5i)T^{2} \) |
| 43 | \( 1 + (0.0159 + 0.0208i)T + (-11.1 + 41.5i)T^{2} \) |
| 47 | \( 1 + (0.938 + 0.541i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (0.240 - 0.581i)T + (-37.4 - 37.4i)T^{2} \) |
| 59 | \( 1 + (10.3 - 1.35i)T + (56.9 - 15.2i)T^{2} \) |
| 61 | \( 1 + (-0.259 + 1.96i)T + (-58.9 - 15.7i)T^{2} \) |
| 67 | \( 1 + (0.569 - 0.742i)T + (-17.3 - 64.7i)T^{2} \) |
| 71 | \( 1 + (-7.83 - 7.83i)T + 71iT^{2} \) |
| 73 | \( 1 + (5.95 - 5.95i)T - 73iT^{2} \) |
| 79 | \( 1 + (-0.917 + 1.58i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (2.87 + 0.378i)T + (80.1 + 21.4i)T^{2} \) |
| 89 | \( 1 + (4.48 - 4.48i)T - 89iT^{2} \) |
| 97 | \( 1 + (7.05 - 12.2i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.63288685646599401577304835584, −10.72758874660979524080326827393, −9.495684344544665343411554450947, −9.267589169979740693135863569548, −8.250488597781771273406921232441, −7.00883428847592077958138014011, −5.50901310610391742878457500826, −4.15529379148300667449232496617, −3.27196187253639927822476054701, −0.968428185588978530457697009495,
1.49806348553457560423036341877, 3.02861924481814379726987059065, 5.35740855034543501411820450034, 6.30234102379022455296838534597, 7.28706632694453116471954274549, 7.66034461814502001779222078275, 9.074228998809026374262708525292, 9.840765275680089949960262833793, 10.99588283428041046955573692228, 11.86421171521093489850830417740
