| L(s) = 1 | + (−0.259 + 1.39i)2-s + (−1.71 + 0.260i)3-s + (−1.86 − 0.720i)4-s + (−0.173 − 1.31i)5-s + (0.0812 − 2.44i)6-s + (0.0444 + 0.165i)7-s + (1.48 − 2.40i)8-s + (2.86 − 0.893i)9-s + (1.87 + 0.100i)10-s + (−0.877 + 0.673i)11-s + (3.38 + 0.747i)12-s + (2.54 − 3.32i)13-s + (−0.242 + 0.0187i)14-s + (0.641 + 2.21i)15-s + (2.96 + 2.68i)16-s + 1.67·17-s + ⋯ |
| L(s) = 1 | + (−0.183 + 0.983i)2-s + (−0.988 + 0.150i)3-s + (−0.932 − 0.360i)4-s + (−0.0776 − 0.590i)5-s + (0.0331 − 0.999i)6-s + (0.0167 + 0.0626i)7-s + (0.525 − 0.850i)8-s + (0.954 − 0.297i)9-s + (0.594 + 0.0317i)10-s + (−0.264 + 0.203i)11-s + (0.976 + 0.215i)12-s + (0.706 − 0.921i)13-s + (−0.0646 + 0.00502i)14-s + (0.165 + 0.571i)15-s + (0.740 + 0.672i)16-s + 0.406·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 - 0.186i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.982 - 0.186i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.756882 + 0.0713302i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.756882 + 0.0713302i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.259 - 1.39i)T \) |
| 3 | \( 1 + (1.71 - 0.260i)T \) |
| good | 5 | \( 1 + (0.173 + 1.31i)T + (-4.82 + 1.29i)T^{2} \) |
| 7 | \( 1 + (-0.0444 - 0.165i)T + (-6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (0.877 - 0.673i)T + (2.84 - 10.6i)T^{2} \) |
| 13 | \( 1 + (-2.54 + 3.32i)T + (-3.36 - 12.5i)T^{2} \) |
| 17 | \( 1 - 1.67T + 17T^{2} \) |
| 19 | \( 1 + (-3.90 + 1.61i)T + (13.4 - 13.4i)T^{2} \) |
| 23 | \( 1 + (-1.65 - 0.443i)T + (19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (6.46 + 0.851i)T + (28.0 + 7.50i)T^{2} \) |
| 31 | \( 1 + (-3.09 + 1.78i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.58 + 8.64i)T + (-26.1 - 26.1i)T^{2} \) |
| 41 | \( 1 + (0.790 - 2.94i)T + (-35.5 - 20.5i)T^{2} \) |
| 43 | \( 1 + (5.37 + 7.00i)T + (-11.1 + 41.5i)T^{2} \) |
| 47 | \( 1 + (0.598 + 0.345i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.43 + 10.7i)T + (-37.4 - 37.4i)T^{2} \) |
| 59 | \( 1 + (-3.24 + 0.427i)T + (56.9 - 15.2i)T^{2} \) |
| 61 | \( 1 + (1.28 - 9.76i)T + (-58.9 - 15.7i)T^{2} \) |
| 67 | \( 1 + (5.81 - 7.57i)T + (-17.3 - 64.7i)T^{2} \) |
| 71 | \( 1 + (-0.107 - 0.107i)T + 71iT^{2} \) |
| 73 | \( 1 + (-3.93 + 3.93i)T - 73iT^{2} \) |
| 79 | \( 1 + (-0.777 + 1.34i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-4.82 - 0.635i)T + (80.1 + 21.4i)T^{2} \) |
| 89 | \( 1 + (10.7 - 10.7i)T - 89iT^{2} \) |
| 97 | \( 1 + (6.90 - 11.9i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.89019126296042858584003864137, −10.75469559764242518246029312657, −9.889417795905908245437865596605, −8.926159832609640055322949401539, −7.83212571880987968879489091389, −6.90973167649607190595565247299, −5.65015571613230783485406090169, −5.18645362179004444303154383041, −3.88643107321076503455553543194, −0.827982617094098833481243066748,
1.37833365279100510673366727204, 3.17974207881909527772334908539, 4.46780741145023075751150330212, 5.64293299987808246143498781937, 6.87500020556484729710790180235, 7.955536376767406460325733282767, 9.268406232052577064806449370057, 10.20157819281589061262731718653, 11.02803230303387692037989899693, 11.55967976909489279478158644807
