| L(s) = 1 | + (−1.39 − 0.236i)2-s + (1.77 − 1.53i)3-s + (1.88 + 0.660i)4-s + (−3.72 + 0.536i)5-s + (−2.83 + 1.72i)6-s + (1.80 − 3.95i)7-s + (−2.47 − 1.36i)8-s + (0.356 − 2.48i)9-s + (5.32 + 0.136i)10-s + (0.791 − 2.69i)11-s + (4.36 − 1.72i)12-s + (−1.31 + 0.602i)13-s + (−3.45 + 5.08i)14-s + (−5.79 + 6.68i)15-s + (3.12 + 2.49i)16-s + (0.426 + 0.274i)17-s + ⋯ |
| L(s) = 1 | + (−0.985 − 0.167i)2-s + (1.02 − 0.887i)3-s + (0.943 + 0.330i)4-s + (−1.66 + 0.239i)5-s + (−1.15 + 0.703i)6-s + (0.682 − 1.49i)7-s + (−0.875 − 0.483i)8-s + (0.118 − 0.826i)9-s + (1.68 + 0.0430i)10-s + (0.238 − 0.812i)11-s + (1.25 − 0.499i)12-s + (−0.365 + 0.167i)13-s + (−0.923 + 1.35i)14-s + (−1.49 + 1.72i)15-s + (0.781 + 0.623i)16-s + (0.103 + 0.0664i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.244 + 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.244 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.510244 - 0.654719i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.510244 - 0.654719i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.39 + 0.236i)T \) |
| 23 | \( 1 + (-4.04 + 2.57i)T \) |
| good | 3 | \( 1 + (-1.77 + 1.53i)T + (0.426 - 2.96i)T^{2} \) |
| 5 | \( 1 + (3.72 - 0.536i)T + (4.79 - 1.40i)T^{2} \) |
| 7 | \( 1 + (-1.80 + 3.95i)T + (-4.58 - 5.29i)T^{2} \) |
| 11 | \( 1 + (-0.791 + 2.69i)T + (-9.25 - 5.94i)T^{2} \) |
| 13 | \( 1 + (1.31 - 0.602i)T + (8.51 - 9.82i)T^{2} \) |
| 17 | \( 1 + (-0.426 - 0.274i)T + (7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (1.09 + 1.69i)T + (-7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (-0.637 + 0.991i)T + (-12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (5.28 - 6.09i)T + (-4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (-1.78 - 0.256i)T + (35.5 + 10.4i)T^{2} \) |
| 41 | \( 1 + (-1.06 - 7.38i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (-0.318 + 0.275i)T + (6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 - 11.5T + 47T^{2} \) |
| 53 | \( 1 + (-8.99 - 4.10i)T + (34.7 + 40.0i)T^{2} \) |
| 59 | \( 1 + (1.14 - 0.524i)T + (38.6 - 44.5i)T^{2} \) |
| 61 | \( 1 + (-6.78 - 5.87i)T + (8.68 + 60.3i)T^{2} \) |
| 67 | \( 1 + (-0.439 - 1.49i)T + (-56.3 + 36.2i)T^{2} \) |
| 71 | \( 1 + (3.81 - 1.11i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (3.83 - 2.46i)T + (30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (4.36 + 9.55i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (-9.43 - 1.35i)T + (79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (7.96 + 9.18i)T + (-12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 + (-0.199 - 1.38i)T + (-93.0 + 27.3i)T^{2} \) |
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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
−12.14691753501689001419297597577, −11.17673832554892950942517769401, −10.59470999652371033830444432748, −8.833035481878164029821331154883, −8.197749814376184518659709564270, −7.33606511425657952778015565659, −7.01739437251957169834949443604, −4.11602438998984778848948228590, −2.98170546855744057490873293396, −0.981613584109564314594969330588,
2.45892038693358315474948179129, 3.90691848289182562568983020548, 5.29784397652168731516197177322, 7.30731053988834516464796706579, 8.128618000307683844892071856822, 8.851780107659473881002823456184, 9.490952260916463506256243620033, 10.84648775935111453860366795454, 11.81511885609027282937056113349, 12.40138404178706666680570836575
