| L(s) = 1 | + (0.157 + 0.432i)2-s + (−0.888 + 1.48i)3-s + (1.36 − 1.14i)4-s + (−0.725 − 4.11i)5-s + (−0.783 − 0.149i)6-s + (2.17 − 1.50i)7-s + (1.51 + 0.871i)8-s + (−1.42 − 2.64i)9-s + (1.66 − 0.961i)10-s + (−1.36 − 0.240i)11-s + (0.492 + 3.05i)12-s + (−1.43 + 3.94i)13-s + (0.992 + 0.706i)14-s + (6.75 + 2.57i)15-s + (0.481 − 2.73i)16-s + (1.18 + 2.06i)17-s + ⋯ |
| L(s) = 1 | + (0.111 + 0.305i)2-s + (−0.512 + 0.858i)3-s + (0.684 − 0.574i)4-s + (−0.324 − 1.83i)5-s + (−0.319 − 0.0612i)6-s + (0.823 − 0.567i)7-s + (0.533 + 0.308i)8-s + (−0.474 − 0.880i)9-s + (0.526 − 0.303i)10-s + (−0.410 − 0.0724i)11-s + (0.142 + 0.882i)12-s + (−0.397 + 1.09i)13-s + (0.265 + 0.188i)14-s + (1.74 + 0.664i)15-s + (0.120 − 0.682i)16-s + (0.288 + 0.499i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.931 + 0.363i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.931 + 0.363i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.20632 - 0.226874i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.20632 - 0.226874i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.888 - 1.48i)T \) |
| 7 | \( 1 + (-2.17 + 1.50i)T \) |
| good | 2 | \( 1 + (-0.157 - 0.432i)T + (-1.53 + 1.28i)T^{2} \) |
| 5 | \( 1 + (0.725 + 4.11i)T + (-4.69 + 1.71i)T^{2} \) |
| 11 | \( 1 + (1.36 + 0.240i)T + (10.3 + 3.76i)T^{2} \) |
| 13 | \( 1 + (1.43 - 3.94i)T + (-9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-1.18 - 2.06i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-4.72 - 2.72i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.01 + 1.21i)T + (-3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (-0.495 - 1.36i)T + (-22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (1.18 + 1.41i)T + (-5.38 + 30.5i)T^{2} \) |
| 37 | \( 1 + (-3.37 - 5.83i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (6.07 + 2.21i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (-0.819 + 4.64i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (-2.18 - 1.83i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 - 11.6iT - 53T^{2} \) |
| 59 | \( 1 + (-1.15 - 6.54i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (1.37 - 1.64i)T + (-10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (-9.25 - 3.36i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (9.79 - 5.65i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (7.69 + 4.44i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (2.14 - 0.779i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-10.9 + 4.00i)T + (63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (-4.23 + 7.34i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (5.77 + 1.01i)T + (91.1 + 33.1i)T^{2} \) |
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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
−12.07989645085724609309168131018, −11.70833496518418042603247800377, −10.54142604586502592350217301063, −9.613309890260015111511218051821, −8.508380909275385944266760137272, −7.42225030320795385739391300411, −5.78801172789346603322911574489, −4.98886022246362121386242860251, −4.20076174245644016375606420852, −1.32131213310530824579739072216,
2.32305440288803489009754620431, 3.15200902425185064646230333680, 5.35224825754222623143699053491, 6.60971633123986465368681790030, 7.54714808909415279785420620973, 7.917887981589640023962069277302, 10.15391939849649765935779074711, 11.10605912218035553021549063912, 11.50427608097385407289618127387, 12.32467176357804439081724288603
