| L(s) = 1 | + (1.62 + 1.36i)2-s + (0.430 + 2.44i)4-s + (−2.52 − 0.917i)5-s + (0.168 − 0.957i)7-s + (−0.508 + 0.880i)8-s + (−2.83 − 4.91i)10-s + (−0.297 + 0.108i)11-s + (−1.15 + 0.973i)13-s + (1.57 − 1.32i)14-s + (2.63 − 0.960i)16-s + (0.587 + 1.01i)17-s + (−3.11 + 5.38i)19-s + (1.15 − 6.55i)20-s + (−0.630 − 0.229i)22-s + (−0.375 − 2.12i)23-s + ⋯ |
| L(s) = 1 | + (1.14 + 0.962i)2-s + (0.215 + 1.22i)4-s + (−1.12 − 0.410i)5-s + (0.0638 − 0.361i)7-s + (−0.179 + 0.311i)8-s + (−0.897 − 1.55i)10-s + (−0.0897 + 0.0326i)11-s + (−0.321 + 0.269i)13-s + (0.421 − 0.353i)14-s + (0.659 − 0.240i)16-s + (0.142 + 0.246i)17-s + (−0.713 + 1.23i)19-s + (0.258 − 1.46i)20-s + (−0.134 − 0.0489i)22-s + (−0.0783 − 0.444i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.558 - 0.829i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.558 - 0.829i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.23885 + 0.659796i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.23885 + 0.659796i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + (-1.62 - 1.36i)T + (0.347 + 1.96i)T^{2} \) |
| 5 | \( 1 + (2.52 + 0.917i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (-0.168 + 0.957i)T + (-6.57 - 2.39i)T^{2} \) |
| 11 | \( 1 + (0.297 - 0.108i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (1.15 - 0.973i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.587 - 1.01i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (3.11 - 5.38i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.375 + 2.12i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-3.37 - 2.83i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (1.50 + 8.54i)T + (-29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (-2.23 - 3.86i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (4.47 - 3.75i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (5.25 - 1.91i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-0.429 + 2.43i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 - 10.8T + 53T^{2} \) |
| 59 | \( 1 + (1.62 + 0.589i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-0.176 + 0.999i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-0.656 + 0.550i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (4.79 + 8.31i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-7.62 + 13.1i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (8.59 + 7.20i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (3.58 + 3.01i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (7.74 - 13.4i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-5.21 + 1.89i)T + (74.3 - 62.3i)T^{2} \) |
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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
−14.73608939961004257307027385860, −13.60214617812699396629547393585, −12.57118518474986973730110368133, −11.78304709076329305515119407364, −10.22043161540038811857623339827, −8.327147249132452417193232130330, −7.47590079156417937225021135284, −6.19662256635009239446625293031, −4.69640221119488074749108962563, −3.79087644557024191915964516651,
2.73169922799128222426771254845, 4.02206801099561134492346145659, 5.29266228537374159111820897485, 7.10445485098415492195622758903, 8.530733050308317175443283904714, 10.32696695861243282350167302515, 11.32681938190540223859830565455, 11.99554850056090211367680382707, 12.94257002574004850007043128460, 14.05014563551942498906646946443
