| L(s) = 1 | + (1.71 + 1.81i)2-s + (−1.10 − 1.33i)3-s + (−0.245 + 4.21i)4-s + (1.39 − 0.163i)5-s + (0.531 − 4.28i)6-s + (−2.85 − 1.43i)7-s + (−4.25 + 3.56i)8-s + (−0.561 + 2.94i)9-s + (2.69 + 2.25i)10-s + (−1.87 − 4.34i)11-s + (5.90 − 4.32i)12-s + (2.07 + 0.491i)13-s + (−2.28 − 7.62i)14-s + (−1.76 − 1.68i)15-s + (−5.36 − 0.627i)16-s + (0.520 − 2.95i)17-s + ⋯ |
| L(s) = 1 | + (1.21 + 1.28i)2-s + (−0.637 − 0.770i)3-s + (−0.122 + 2.10i)4-s + (0.625 − 0.0730i)5-s + (0.216 − 1.75i)6-s + (−1.07 − 0.541i)7-s + (−1.50 + 1.26i)8-s + (−0.187 + 0.982i)9-s + (0.850 + 0.713i)10-s + (−0.565 − 1.31i)11-s + (1.70 − 1.24i)12-s + (0.575 + 0.136i)13-s + (−0.610 − 2.03i)14-s + (−0.454 − 0.435i)15-s + (−1.34 − 0.156i)16-s + (0.126 − 0.715i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.456 - 0.889i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.456 - 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.16737 + 0.713530i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.16737 + 0.713530i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.10 + 1.33i)T \) |
| good | 2 | \( 1 + (-1.71 - 1.81i)T + (-0.116 + 1.99i)T^{2} \) |
| 5 | \( 1 + (-1.39 + 0.163i)T + (4.86 - 1.15i)T^{2} \) |
| 7 | \( 1 + (2.85 + 1.43i)T + (4.18 + 5.61i)T^{2} \) |
| 11 | \( 1 + (1.87 + 4.34i)T + (-7.54 + 8.00i)T^{2} \) |
| 13 | \( 1 + (-2.07 - 0.491i)T + (11.6 + 5.83i)T^{2} \) |
| 17 | \( 1 + (-0.520 + 2.95i)T + (-15.9 - 5.81i)T^{2} \) |
| 19 | \( 1 + (-1.23 - 6.99i)T + (-17.8 + 6.49i)T^{2} \) |
| 23 | \( 1 + (2.70 - 1.35i)T + (13.7 - 18.4i)T^{2} \) |
| 29 | \( 1 + (0.631 - 2.10i)T + (-24.2 - 15.9i)T^{2} \) |
| 31 | \( 1 + (-3.14 - 2.06i)T + (12.2 + 28.4i)T^{2} \) |
| 37 | \( 1 + (6.31 - 2.29i)T + (28.3 - 23.7i)T^{2} \) |
| 41 | \( 1 + (-2.90 + 3.08i)T + (-2.38 - 40.9i)T^{2} \) |
| 43 | \( 1 + (-0.474 + 0.636i)T + (-12.3 - 41.1i)T^{2} \) |
| 47 | \( 1 + (-4.83 + 3.17i)T + (18.6 - 43.1i)T^{2} \) |
| 53 | \( 1 + (-4.83 + 8.37i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (0.636 - 1.47i)T + (-40.4 - 42.9i)T^{2} \) |
| 61 | \( 1 + (-0.0674 - 1.15i)T + (-60.5 + 7.08i)T^{2} \) |
| 67 | \( 1 + (-1.71 - 5.73i)T + (-55.9 + 36.8i)T^{2} \) |
| 71 | \( 1 + (-2.74 - 2.30i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (-8.55 + 7.18i)T + (12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (3.21 + 3.41i)T + (-4.59 + 78.8i)T^{2} \) |
| 83 | \( 1 + (10.2 + 10.8i)T + (-4.82 + 82.8i)T^{2} \) |
| 89 | \( 1 + (-3.17 + 2.66i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (12.5 + 1.46i)T + (94.3 + 22.3i)T^{2} \) |
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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
−13.98544343885493143284447963027, −13.72368004233707833286103588710, −12.89969405783094585332028520599, −11.85091882340504487136018290947, −10.26351332649326099063182998062, −8.303448453360563535564442278533, −7.13578468695699420125264122555, −6.07518762030627618367331730304, −5.51489658269140328932307128314, −3.50257998028236303039454351837,
2.61001728858544505879515016203, 4.14306965967275002060377765318, 5.40790265638291002490333542330, 6.36782004037908675797598088669, 9.416177819326844269001826492204, 10.05993030315965215760288713533, 10.98400444949323542686400492333, 12.19559147052195986771462345658, 12.83719982348967830336605379137, 13.83943042211186480092798603016
