| 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
−12.32467176357804439081724288603, −11.50427608097385407289618127387, −11.10605912218035553021549063912, −10.15391939849649765935779074711, −7.917887981589640023962069277302, −7.54714808909415279785420620973, −6.60971633123986465368681790030, −5.35224825754222623143699053491, −3.15200902425185064646230333680, −2.32305440288803489009754620431,
1.32131213310530824579739072216, 4.20076174245644016375606420852, 4.98886022246362121386242860251, 5.78801172789346603322911574489, 7.42225030320795385739391300411, 8.508380909275385944266760137272, 9.613309890260015111511218051821, 10.54142604586502592350217301063, 11.70833496518418042603247800377, 12.07989645085724609309168131018
