L(s) = 1 | + 1.73·3-s + (−0.848 − 0.848i)5-s + (5.42 − 5.42i)7-s + 2.99·9-s + (3.83 − 3.83i)11-s + (12.9 + 1.59i)13-s + (−1.46 − 1.46i)15-s + 2.95i·17-s + (−0.754 − 0.754i)19-s + (9.40 − 9.40i)21-s + 19.1i·23-s − 23.5i·25-s + 5.19·27-s − 34.3·29-s + (14.8 + 14.8i)31-s + ⋯ |
L(s) = 1 | + 0.577·3-s + (−0.169 − 0.169i)5-s + (0.775 − 0.775i)7-s + 0.333·9-s + (0.348 − 0.348i)11-s + (0.992 + 0.122i)13-s + (−0.0979 − 0.0979i)15-s + 0.173i·17-s + (−0.0397 − 0.0397i)19-s + (0.447 − 0.447i)21-s + 0.831i·23-s − 0.942i·25-s + 0.192·27-s − 1.18·29-s + (0.477 + 0.477i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.914 + 0.404i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.914 + 0.404i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.81874 - 0.384763i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.81874 - 0.384763i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - 1.73T \) |
| 13 | \( 1 + (-12.9 - 1.59i)T \) |
good | 5 | \( 1 + (0.848 + 0.848i)T + 25iT^{2} \) |
| 7 | \( 1 + (-5.42 + 5.42i)T - 49iT^{2} \) |
| 11 | \( 1 + (-3.83 + 3.83i)T - 121iT^{2} \) |
| 17 | \( 1 - 2.95iT - 289T^{2} \) |
| 19 | \( 1 + (0.754 + 0.754i)T + 361iT^{2} \) |
| 23 | \( 1 - 19.1iT - 529T^{2} \) |
| 29 | \( 1 + 34.3T + 841T^{2} \) |
| 31 | \( 1 + (-14.8 - 14.8i)T + 961iT^{2} \) |
| 37 | \( 1 + (-2.18 + 2.18i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 + (12.9 + 12.9i)T + 1.68e3iT^{2} \) |
| 43 | \( 1 - 22.0iT - 1.84e3T^{2} \) |
| 47 | \( 1 + (52.5 - 52.5i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + 101.T + 2.80e3T^{2} \) |
| 59 | \( 1 + (32.0 - 32.0i)T - 3.48e3iT^{2} \) |
| 61 | \( 1 + 43.7T + 3.72e3T^{2} \) |
| 67 | \( 1 + (-12.5 - 12.5i)T + 4.48e3iT^{2} \) |
| 71 | \( 1 + (-41.2 - 41.2i)T + 5.04e3iT^{2} \) |
| 73 | \( 1 + (-26.4 + 26.4i)T - 5.32e3iT^{2} \) |
| 79 | \( 1 + 27.4T + 6.24e3T^{2} \) |
| 83 | \( 1 + (-96.6 - 96.6i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 + (-98.1 + 98.1i)T - 7.92e3iT^{2} \) |
| 97 | \( 1 + (26.1 + 26.1i)T + 9.40e3iT^{2} \) |
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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
−12.79401627727492986060217676249, −11.48386012605369603596361163865, −10.74308668961063725459932178815, −9.462355273078906518723263599575, −8.390582215667555441432990509074, −7.60909445086515288776064223964, −6.24454258447296375648918593442, −4.60430464458041502922492186939, −3.49495439572271045802482207057, −1.43956879458254923958440913599,
1.87490308967734872209532679582, 3.50369228236151632686274322922, 4.95498992024679537797426143501, 6.36360226574454310302948487208, 7.73070210027502395640880943157, 8.609675292523212397934507099455, 9.522102683585003077007313687381, 10.89119054795782602938878867050, 11.71434751655851616232543072959, 12.81881661445438478408023226271