L(s) = 1 | + (1.21 + 2.09i)2-s + (−0.689 + 1.58i)3-s + (−1.93 + 3.35i)4-s + (−0.5 + 0.866i)5-s + (−4.17 + 0.478i)6-s + (0.905 + 1.56i)7-s − 4.54·8-s + (−2.04 − 2.19i)9-s − 2.42·10-s + (−0.367 − 0.635i)11-s + (−3.99 − 5.39i)12-s + (−0.5 + 0.866i)13-s + (−2.19 + 3.80i)14-s + (−1.03 − 1.39i)15-s + (−1.63 − 2.82i)16-s + 4.77·17-s + ⋯ |
L(s) = 1 | + (0.856 + 1.48i)2-s + (−0.397 + 0.917i)3-s + (−0.968 + 1.67i)4-s + (−0.223 + 0.387i)5-s + (−1.70 + 0.195i)6-s + (0.342 + 0.592i)7-s − 1.60·8-s + (−0.683 − 0.730i)9-s − 0.766·10-s + (−0.110 − 0.191i)11-s + (−1.15 − 1.55i)12-s + (−0.138 + 0.240i)13-s + (−0.586 + 1.01i)14-s + (−0.266 − 0.359i)15-s + (−0.407 − 0.706i)16-s + 1.15·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.600 + 0.799i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.600 + 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.746160 - 1.49340i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.746160 - 1.49340i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.689 - 1.58i)T \) |
| 5 | \( 1 + (0.5 - 0.866i)T \) |
| 13 | \( 1 + (0.5 - 0.866i)T \) |
good | 2 | \( 1 + (-1.21 - 2.09i)T + (-1 + 1.73i)T^{2} \) |
| 7 | \( 1 + (-0.905 - 1.56i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (0.367 + 0.635i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 - 4.77T + 17T^{2} \) |
| 19 | \( 1 + 4.07T + 19T^{2} \) |
| 23 | \( 1 + (0.336 - 0.582i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.25 - 2.17i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-3.27 + 5.67i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 6.08T + 37T^{2} \) |
| 41 | \( 1 + (1.56 - 2.70i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-4.46 - 7.72i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (1.78 + 3.09i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 12.8T + 53T^{2} \) |
| 59 | \( 1 + (6.66 - 11.5i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.42 - 4.19i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.43 + 2.48i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 9.86T + 71T^{2} \) |
| 73 | \( 1 - 11.8T + 73T^{2} \) |
| 79 | \( 1 + (-2.05 - 3.56i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (6.35 + 11.0i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 5.47T + 89T^{2} \) |
| 97 | \( 1 + (-0.851 - 1.47i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−11.40473251115887468974553066690, −10.37585094053189537421523238374, −9.331705534186036000218692809418, −8.348355154992968284292123672171, −7.63886837784951518007420891175, −6.37723176482088602165082502440, −5.86434304312239102638973848269, −4.89877919024502496337130966809, −4.14718630502662259904462718093, −3.01802222892111318661480984153,
0.78246052598545691982104794144, 1.89934266470487007404717207214, 3.13982729735380299043298368077, 4.38966805167842899797963014337, 5.16512696027714980245781438678, 6.20387727400462797712535229379, 7.49604955079213789843667344939, 8.315931692263272946007963886989, 9.672885815866695018409703277800, 10.60176251566314351261972540080