L(s) = 1 | + (−2.08 + 2.08i)2-s − 4.73i·4-s + (−0.137 + 4.99i)5-s + (−1.87 + 1.87i)7-s + (1.53 + 1.53i)8-s + (−10.1 − 10.7i)10-s − 2.70·11-s + (−2.37 − 2.37i)13-s − 7.81i·14-s + 12.5·16-s + (−16.3 + 16.3i)17-s + 9.18i·19-s + (23.6 + 0.649i)20-s + (5.64 − 5.64i)22-s + (−21.4 − 21.4i)23-s + ⋯ |
L(s) = 1 | + (−1.04 + 1.04i)2-s − 1.18i·4-s + (−0.0274 + 0.999i)5-s + (−0.267 + 0.267i)7-s + (0.191 + 0.191i)8-s + (−1.01 − 1.07i)10-s − 0.245·11-s + (−0.183 − 0.183i)13-s − 0.558i·14-s + 0.782·16-s + (−0.963 + 0.963i)17-s + 0.483i·19-s + (1.18 + 0.0324i)20-s + (0.256 − 0.256i)22-s + (−0.932 − 0.932i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.202 + 0.979i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.202 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0269267 - 0.0219176i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0269267 - 0.0219176i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (0.137 - 4.99i)T \) |
| 7 | \( 1 + (1.87 - 1.87i)T \) |
good | 2 | \( 1 + (2.08 - 2.08i)T - 4iT^{2} \) |
| 11 | \( 1 + 2.70T + 121T^{2} \) |
| 13 | \( 1 + (2.37 + 2.37i)T + 169iT^{2} \) |
| 17 | \( 1 + (16.3 - 16.3i)T - 289iT^{2} \) |
| 19 | \( 1 - 9.18iT - 361T^{2} \) |
| 23 | \( 1 + (21.4 + 21.4i)T + 529iT^{2} \) |
| 29 | \( 1 + 52.3iT - 841T^{2} \) |
| 31 | \( 1 + 5.01T + 961T^{2} \) |
| 37 | \( 1 + (-23.2 + 23.2i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 - 60.5T + 1.68e3T^{2} \) |
| 43 | \( 1 + (8.78 + 8.78i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (2.24 - 2.24i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (-25.6 - 25.6i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 + 100. iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 82.1T + 3.72e3T^{2} \) |
| 67 | \( 1 + (65.1 - 65.1i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 - 22.8T + 5.04e3T^{2} \) |
| 73 | \( 1 + (5.38 + 5.38i)T + 5.32e3iT^{2} \) |
| 79 | \( 1 + 117. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (85.5 + 85.5i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 - 119. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (55.1 - 55.1i)T - 9.40e3iT^{2} \) |
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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
−12.06437247627456901682692757739, −10.82464534564487061336271078504, −10.13529227550120066984113863892, −9.264703468794996500728430287010, −8.169436292215725104775094126964, −7.55078624509756233115266067561, −6.35974139912342358813072664200, −5.97760976662915448505051080333, −4.01744104294645452345796229371, −2.39664210826052631645385571014,
0.02362835841349120692011332710, 1.41875694949684677846391194626, 2.81704238438118240782423541759, 4.32433456769907039550751301525, 5.57168601520163898526394899807, 7.17348138682535905905089868168, 8.221776878405126657126821081975, 9.167036636650116527691358024036, 9.587052778113531437990461517389, 10.70075426700684905534670938794