L(s) = 1 | + (0.118 − 0.673i)2-s + (0.500 + 0.181i)4-s + (−0.342 − 0.939i)5-s + (0.524 − 0.907i)8-s + (−0.673 + 0.118i)10-s + (−0.141 − 0.118i)16-s + (0.342 + 0.0603i)17-s + (−0.766 − 0.642i)19-s − 0.532i·20-s + (−0.342 + 0.939i)23-s + (−0.766 + 0.642i)25-s + (1.11 − 0.642i)31-s + (0.705 − 0.592i)32-s + (0.0812 − 0.223i)34-s + (−0.524 + 0.439i)38-s + ⋯ |
L(s) = 1 | + (0.118 − 0.673i)2-s + (0.500 + 0.181i)4-s + (−0.342 − 0.939i)5-s + (0.524 − 0.907i)8-s + (−0.673 + 0.118i)10-s + (−0.141 − 0.118i)16-s + (0.342 + 0.0603i)17-s + (−0.766 − 0.642i)19-s − 0.532i·20-s + (−0.342 + 0.939i)23-s + (−0.766 + 0.642i)25-s + (1.11 − 0.642i)31-s + (0.705 − 0.592i)32-s + (0.0812 − 0.223i)34-s + (−0.524 + 0.439i)38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.150 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.150 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.166504806\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.166504806\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (0.342 + 0.939i)T \) |
| 19 | \( 1 + (0.766 + 0.642i)T \) |
good | 2 | \( 1 + (-0.118 + 0.673i)T + (-0.939 - 0.342i)T^{2} \) |
| 7 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (0.173 - 0.984i)T^{2} \) |
| 17 | \( 1 + (-0.342 - 0.0603i)T + (0.939 + 0.342i)T^{2} \) |
| 23 | \( 1 + (0.342 - 0.939i)T + (-0.766 - 0.642i)T^{2} \) |
| 29 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 31 | \( 1 + (-1.11 + 0.642i)T + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 43 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 47 | \( 1 + (1.85 - 0.326i)T + (0.939 - 0.342i)T^{2} \) |
| 53 | \( 1 + (-1.85 - 0.673i)T + (0.766 + 0.642i)T^{2} \) |
| 59 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 61 | \( 1 + (-0.939 - 0.342i)T + (0.766 + 0.642i)T^{2} \) |
| 67 | \( 1 + (-0.939 + 0.342i)T^{2} \) |
| 71 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 73 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 79 | \( 1 + (1.11 - 1.32i)T + (-0.173 - 0.984i)T^{2} \) |
| 83 | \( 1 + (1.32 - 0.766i)T + (0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 97 | \( 1 + (-0.939 - 0.342i)T^{2} \) |
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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
−10.21237122862832331757294958278, −9.534781220818219199867878963363, −8.477961737760453039500265635332, −7.75678847619785463234315705656, −6.81156550100759589273502682879, −5.74501540054493952505229915139, −4.57607218152830034867307652390, −3.78039074494760301228968582711, −2.58133090911437607213258166813, −1.30194884373983814357289418389,
2.03219757161604765454578288860, 3.14136757211673628466840272482, 4.38449293040011317977396685450, 5.54946441385440889157976111924, 6.49150378378436827623040322905, 6.90947593548681170527051244915, 7.964984214405840783881739457646, 8.488305304276225878787749467293, 10.13116231816327959567218878412, 10.36725321867941891361159340777