L(s) = 1 | + (−1.02 − 0.592i)2-s + (−0.296 − 0.514i)4-s + (1.41 + 2.45i)5-s + (−0.387 + 2.61i)7-s + 3.07i·8-s − 3.36i·10-s + (−0.136 − 0.0789i)11-s + (3.41 − 1.97i)13-s + (1.95 − 2.45i)14-s + (1.23 − 2.13i)16-s + 4.14·17-s + 6.33i·19-s + (0.842 − 1.45i)20-s + (0.0935 + 0.162i)22-s + (−0.472 + 0.273i)23-s + ⋯ |
L(s) = 1 | + (−0.726 − 0.419i)2-s + (−0.148 − 0.257i)4-s + (0.634 + 1.09i)5-s + (−0.146 + 0.989i)7-s + 1.08i·8-s − 1.06i·10-s + (−0.0412 − 0.0237i)11-s + (0.947 − 0.546i)13-s + (0.521 − 0.656i)14-s + (0.307 − 0.532i)16-s + 1.00·17-s + 1.45i·19-s + (0.188 − 0.326i)20-s + (0.0199 + 0.0345i)22-s + (−0.0986 + 0.0569i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.904 - 0.426i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.904 - 0.426i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.814337 + 0.182490i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.814337 + 0.182490i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (0.387 - 2.61i)T \) |
good | 2 | \( 1 + (1.02 + 0.592i)T + (1 + 1.73i)T^{2} \) |
| 5 | \( 1 + (-1.41 - 2.45i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (0.136 + 0.0789i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-3.41 + 1.97i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 4.14T + 17T^{2} \) |
| 19 | \( 1 - 6.33iT - 19T^{2} \) |
| 23 | \( 1 + (0.472 - 0.273i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (4.02 + 2.32i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (0.112 - 0.0647i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 2.46T + 37T^{2} \) |
| 41 | \( 1 + (1.99 + 3.45i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.28 + 5.68i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (4.33 - 7.50i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 2.60iT - 53T^{2} \) |
| 59 | \( 1 + (-1.80 - 3.12i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.91 - 1.68i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (0.663 + 1.14i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 0.409iT - 71T^{2} \) |
| 73 | \( 1 + 15.0iT - 73T^{2} \) |
| 79 | \( 1 + (2.16 - 3.74i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-3.22 + 5.58i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 5.05T + 89T^{2} \) |
| 97 | \( 1 + (-2.18 - 1.26i)T + (48.5 + 84.0i)T^{2} \) |
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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
−12.43933549698999179051398366543, −11.37681275996531347841757861829, −10.42864420022306771420592508745, −9.883454601855530986434059236476, −8.819660310715993346431449986650, −7.80607823382111515250238665348, −6.10874975306170576459159328006, −5.57250785424656302056517814957, −3.27288882154243200825932417482, −1.89149283792605881726851118836,
1.08014189312548087383128657237, 3.69752586245828272573291308311, 4.95960669514515008917334947596, 6.47262636646649577358634480709, 7.51665778199728448137744382952, 8.606739627649458207924499045286, 9.307027674696705628011271726600, 10.16809721927531272367349822706, 11.45254174544128942784554140816, 12.87410123426954087965499771547