| L(s) = 1 | + (0.716 − 1.21i)2-s + (−0.325 + 1.70i)3-s + (−0.972 − 1.74i)4-s + (−0.471 − 3.58i)5-s + (1.84 + 1.61i)6-s + (−0.240 − 0.898i)7-s + (−2.82 − 0.0662i)8-s + (−2.78 − 1.10i)9-s + (−4.70 − 1.99i)10-s + (−3.26 + 2.50i)11-s + (3.28 − 1.08i)12-s + (3.99 − 5.21i)13-s + (−1.26 − 0.350i)14-s + (6.24 + 0.365i)15-s + (−2.10 + 3.39i)16-s + 0.549·17-s + ⋯ |
| L(s) = 1 | + (0.506 − 0.862i)2-s + (−0.188 + 0.982i)3-s + (−0.486 − 0.873i)4-s + (−0.210 − 1.60i)5-s + (0.751 + 0.659i)6-s + (−0.0909 − 0.339i)7-s + (−0.999 − 0.0234i)8-s + (−0.929 − 0.369i)9-s + (−1.48 − 0.629i)10-s + (−0.985 + 0.756i)11-s + (0.949 − 0.313i)12-s + (1.10 − 1.44i)13-s + (−0.338 − 0.0936i)14-s + (1.61 + 0.0943i)15-s + (−0.526 + 0.849i)16-s + 0.133·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.611 + 0.791i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.611 + 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.532111 - 1.08334i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.532111 - 1.08334i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.716 + 1.21i)T \) |
| 3 | \( 1 + (0.325 - 1.70i)T \) |
| good | 5 | \( 1 + (0.471 + 3.58i)T + (-4.82 + 1.29i)T^{2} \) |
| 7 | \( 1 + (0.240 + 0.898i)T + (-6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (3.26 - 2.50i)T + (2.84 - 10.6i)T^{2} \) |
| 13 | \( 1 + (-3.99 + 5.21i)T + (-3.36 - 12.5i)T^{2} \) |
| 17 | \( 1 - 0.549T + 17T^{2} \) |
| 19 | \( 1 + (-5.33 + 2.20i)T + (13.4 - 13.4i)T^{2} \) |
| 23 | \( 1 + (3.35 + 0.899i)T + (19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (-5.25 - 0.692i)T + (28.0 + 7.50i)T^{2} \) |
| 31 | \( 1 + (-0.693 + 0.400i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (0.929 - 2.24i)T + (-26.1 - 26.1i)T^{2} \) |
| 41 | \( 1 + (0.841 - 3.13i)T + (-35.5 - 20.5i)T^{2} \) |
| 43 | \( 1 + (-4.28 - 5.58i)T + (-11.1 + 41.5i)T^{2} \) |
| 47 | \( 1 + (-1.79 - 1.03i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.71 + 4.14i)T + (-37.4 - 37.4i)T^{2} \) |
| 59 | \( 1 + (-2.13 + 0.281i)T + (56.9 - 15.2i)T^{2} \) |
| 61 | \( 1 + (1.27 - 9.71i)T + (-58.9 - 15.7i)T^{2} \) |
| 67 | \( 1 + (-6.45 + 8.41i)T + (-17.3 - 64.7i)T^{2} \) |
| 71 | \( 1 + (10.4 + 10.4i)T + 71iT^{2} \) |
| 73 | \( 1 + (8.18 - 8.18i)T - 73iT^{2} \) |
| 79 | \( 1 + (0.656 - 1.13i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-7.51 - 0.989i)T + (80.1 + 21.4i)T^{2} \) |
| 89 | \( 1 + (3.96 - 3.96i)T - 89iT^{2} \) |
| 97 | \( 1 + (-6.94 + 12.0i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.53980362685809987309838907923, −10.44687352628942510879091761727, −9.899263261058007377734552225988, −8.865729553211589747902062508119, −8.018714570446380201323059091333, −5.79179835903273853860311425223, −5.08107323613736494127550030499, −4.30494159969741839545783052738, −3.08060969490116475745717165347, −0.818648553868020850644984597995,
2.61229480701141332226937147727, 3.67790128056213950874395023016, 5.62304022740205685905640822364, 6.28617975845111917001403423795, 7.13883451327757022351667052923, 7.893321778634245084965828684450, 8.897403199627121486554480951361, 10.51080470248244236578484307135, 11.52898109930897245320530831151, 12.06235488965714023674227573904
