| L(s) = 1 | + (0.0549 + 1.41i)2-s + (1.56 − 1.35i)3-s + (−1.99 + 0.155i)4-s + (1.30 − 0.188i)5-s + (1.99 + 2.13i)6-s + (1.93 − 4.22i)7-s + (−0.328 − 2.80i)8-s + (0.180 − 1.25i)9-s + (0.337 + 1.83i)10-s + (−0.917 + 3.12i)11-s + (−2.90 + 2.94i)12-s + (−4.99 + 2.28i)13-s + (6.08 + 2.49i)14-s + (1.78 − 2.06i)15-s + (3.95 − 0.619i)16-s + (3.05 + 1.96i)17-s + ⋯ |
| L(s) = 1 | + (0.0388 + 0.999i)2-s + (0.901 − 0.781i)3-s + (−0.996 + 0.0776i)4-s + (0.584 − 0.0840i)5-s + (0.815 + 0.870i)6-s + (0.729 − 1.59i)7-s + (−0.116 − 0.993i)8-s + (0.0602 − 0.419i)9-s + (0.106 + 0.581i)10-s + (−0.276 + 0.941i)11-s + (−0.838 + 0.849i)12-s + (−1.38 + 0.632i)13-s + (1.62 + 0.667i)14-s + (0.461 − 0.532i)15-s + (0.987 − 0.154i)16-s + (0.741 + 0.476i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.952 - 0.305i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.952 - 0.305i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.50834 + 0.236040i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.50834 + 0.236040i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.0549 - 1.41i)T \) |
| 23 | \( 1 + (4.20 - 2.30i)T \) |
| good | 3 | \( 1 + (-1.56 + 1.35i)T + (0.426 - 2.96i)T^{2} \) |
| 5 | \( 1 + (-1.30 + 0.188i)T + (4.79 - 1.40i)T^{2} \) |
| 7 | \( 1 + (-1.93 + 4.22i)T + (-4.58 - 5.29i)T^{2} \) |
| 11 | \( 1 + (0.917 - 3.12i)T + (-9.25 - 5.94i)T^{2} \) |
| 13 | \( 1 + (4.99 - 2.28i)T + (8.51 - 9.82i)T^{2} \) |
| 17 | \( 1 + (-3.05 - 1.96i)T + (7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (-1.11 - 1.73i)T + (-7.89 + 17.2i)T^{2} \) |
| 29 | \( 1 + (0.0805 - 0.125i)T + (-12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (-5.26 + 6.07i)T + (-4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (6.57 + 0.945i)T + (35.5 + 10.4i)T^{2} \) |
| 41 | \( 1 + (0.661 + 4.60i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (4.20 - 3.64i)T + (6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 - 2.17T + 47T^{2} \) |
| 53 | \( 1 + (1.56 + 0.715i)T + (34.7 + 40.0i)T^{2} \) |
| 59 | \( 1 + (-10.3 + 4.71i)T + (38.6 - 44.5i)T^{2} \) |
| 61 | \( 1 + (-0.788 - 0.683i)T + (8.68 + 60.3i)T^{2} \) |
| 67 | \( 1 + (1.96 + 6.68i)T + (-56.3 + 36.2i)T^{2} \) |
| 71 | \( 1 + (-4.09 + 1.20i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (5.31 - 3.41i)T + (30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (-3.36 - 7.37i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (-3.79 - 0.545i)T + (79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (5.49 + 6.34i)T + (-12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 + (-2.47 - 17.2i)T + (-93.0 + 27.3i)T^{2} \) |
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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
−13.12689515308676582851796547657, −12.08738747599668364930989458305, −10.18745618440756934362216597117, −9.636024086339282250251673213531, −8.056735832622834849502628517768, −7.60191994879877007253767983778, −6.84688959636374389375122223066, −5.18692664979833130852998198994, −4.01088502958525951979002210435, −1.80232905808029817814668872454,
2.36821447514302749993815769366, 3.11488870800183240244803961484, 4.87195490729449040470747195852, 5.66662552735158118526329003674, 8.143595524445108819031492678592, 8.745417702757169641582153210102, 9.716311512037980165365007431843, 10.31370665940268986569239065180, 11.76359248449388923406812130780, 12.26885852888743804711012240447
