| L(s) = 1 | + (−0.681 − 1.87i)2-s + (−0.917 + 1.46i)3-s + (−1.50 + 1.26i)4-s + (−0.291 − 1.65i)5-s + (3.37 + 0.716i)6-s + (−2.47 + 0.940i)7-s + (−0.0549 − 0.0317i)8-s + (−1.31 − 2.69i)9-s + (−2.89 + 1.67i)10-s + (−3.13 − 0.553i)11-s + (−0.475 − 3.37i)12-s + (−0.877 + 2.41i)13-s + (3.44 + 3.98i)14-s + (2.69 + 1.08i)15-s + (−0.705 + 4.00i)16-s + (−0.934 − 1.61i)17-s + ⋯ |
| L(s) = 1 | + (−0.481 − 1.32i)2-s + (−0.529 + 0.848i)3-s + (−0.753 + 0.632i)4-s + (−0.130 − 0.739i)5-s + (1.37 + 0.292i)6-s + (−0.934 + 0.355i)7-s + (−0.0194 − 0.0112i)8-s + (−0.438 − 0.898i)9-s + (−0.916 + 0.529i)10-s + (−0.946 − 0.166i)11-s + (−0.137 − 0.974i)12-s + (−0.243 + 0.668i)13-s + (0.920 + 1.06i)14-s + (0.696 + 0.281i)15-s + (−0.176 + 1.00i)16-s + (−0.226 − 0.392i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.783 - 0.621i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.783 - 0.621i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0597880 + 0.171423i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0597880 + 0.171423i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.917 - 1.46i)T \) |
| 7 | \( 1 + (2.47 - 0.940i)T \) |
| good | 2 | \( 1 + (0.681 + 1.87i)T + (-1.53 + 1.28i)T^{2} \) |
| 5 | \( 1 + (0.291 + 1.65i)T + (-4.69 + 1.71i)T^{2} \) |
| 11 | \( 1 + (3.13 + 0.553i)T + (10.3 + 3.76i)T^{2} \) |
| 13 | \( 1 + (0.877 - 2.41i)T + (-9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (0.934 + 1.61i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (5.71 + 3.30i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.62 - 5.50i)T + (-3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (3.25 + 8.94i)T + (-22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (0.942 + 1.12i)T + (-5.38 + 30.5i)T^{2} \) |
| 37 | \( 1 + (0.322 + 0.558i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (0.477 + 0.173i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (0.197 - 1.12i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (8.83 + 7.41i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 - 1.23iT - 53T^{2} \) |
| 59 | \( 1 + (-0.0973 - 0.552i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (-4.58 + 5.46i)T + (-10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (-8.34 - 3.03i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (5.31 - 3.07i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-5.75 - 3.32i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.814 + 0.296i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (7.67 - 2.79i)T + (63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (3.77 - 6.54i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (9.45 + 1.66i)T + (91.1 + 33.1i)T^{2} \) |
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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
−11.66501717474169704972266851509, −11.06793016713207465430897923699, −9.976495095684604300516319139790, −9.354800729803579293368924215169, −8.581780030631774605070113369109, −6.57729111468772739092936813128, −5.22107647937632725803358852848, −3.96326940827278738292783441427, −2.60885126742765460759018939348, −0.18628116788880523832285874561,
2.86924845987549346133030724034, 5.18618973228667074884006782161, 6.35665491077128490895435803626, 6.89911880511410352932577286211, 7.76439772374623046387117435903, 8.723920310097604464203939643061, 10.31186142275917453680545287827, 10.91773027912428049056119241645, 12.61540552305374943742826740361, 12.96318480164419451661297090516
