L(s) = 1 | + (−4.46 − 2.57i)5-s + (−5.25 + 3.03i)11-s + 12.5·13-s + (14.9 − 8.64i)17-s + (−12.9 + 22.4i)19-s + (−2.09 − 1.20i)23-s + (0.791 + 1.37i)25-s − 55.8i·29-s + (7.64 + 13.2i)31-s + (11.8 − 20.5i)37-s + 15.4i·41-s + 27.7·43-s + (−24.6 − 14.2i)47-s + (40.4 − 23.3i)53-s + 31.2·55-s + ⋯ |
L(s) = 1 | + (−0.893 − 0.515i)5-s + (−0.477 + 0.275i)11-s + 0.967·13-s + (0.881 − 0.508i)17-s + (−0.680 + 1.17i)19-s + (−0.0909 − 0.0525i)23-s + (0.0316 + 0.0548i)25-s − 1.92i·29-s + (0.246 + 0.427i)31-s + (0.320 − 0.555i)37-s + 0.377i·41-s + 0.645·43-s + (−0.524 − 0.302i)47-s + (0.762 − 0.440i)53-s + 0.568·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.986 + 0.160i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.986 + 0.160i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3673837372\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3673837372\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (4.46 + 2.57i)T + (12.5 + 21.6i)T^{2} \) |
| 11 | \( 1 + (5.25 - 3.03i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 - 12.5T + 169T^{2} \) |
| 17 | \( 1 + (-14.9 + 8.64i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (12.9 - 22.4i)T + (-180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (2.09 + 1.20i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + 55.8iT - 841T^{2} \) |
| 31 | \( 1 + (-7.64 - 13.2i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (-11.8 + 20.5i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 - 15.4iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 27.7T + 1.84e3T^{2} \) |
| 47 | \( 1 + (24.6 + 14.2i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-40.4 + 23.3i)T + (1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (69.8 - 40.3i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (31.3 - 54.3i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (58.0 + 100. i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 49.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (-15.3 - 26.5i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (71.3 - 123. i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + 28.5iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (145. + 84.1i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + 113.T + 9.40e3T^{2} \) |
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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
−8.474526380588042230436833614569, −8.058852745210814645501394334428, −7.38702475800928048145520353662, −6.21305656869450397200860367494, −5.54604013619349664400647823198, −4.37466237812580506309176041611, −3.88302385570058270480195701428, −2.70401793885347107763848699332, −1.33517757274874330546604158311, −0.10589148357740278200294727633,
1.31562163329042823133068513089, 2.83449135857306516574452887261, 3.52802626670471235810033564477, 4.43439090716146232754458845732, 5.47553544966923233618008117361, 6.36807466442938443461636707053, 7.17500446398064162821973334725, 7.923718414895533153003375873653, 8.585985029175352713288390890798, 9.373389463182017242449990033210