L(s) = 1 | + (0.212 + 0.873i)2-s + (−0.327 − 0.945i)3-s + (1.05 − 0.545i)4-s + (−3.10 + 1.24i)5-s + (0.756 − 0.486i)6-s + (−2.28 − 1.32i)7-s + (1.87 + 2.16i)8-s + (−0.786 + 0.618i)9-s + (−1.74 − 2.44i)10-s + (−0.745 + 3.07i)11-s + (−0.862 − 0.822i)12-s + (−1.43 + 3.13i)13-s + (0.677 − 2.28i)14-s + (2.18 + 2.52i)15-s + (−0.115 + 0.161i)16-s + (0.290 + 6.10i)17-s + ⋯ |
L(s) = 1 | + (0.149 + 0.617i)2-s + (−0.188 − 0.545i)3-s + (0.529 − 0.272i)4-s + (−1.38 + 0.555i)5-s + (0.308 − 0.198i)6-s + (−0.864 − 0.502i)7-s + (0.664 + 0.766i)8-s + (−0.262 + 0.206i)9-s + (−0.551 − 0.774i)10-s + (−0.224 + 0.926i)11-s + (−0.248 − 0.237i)12-s + (−0.396 + 0.868i)13-s + (0.180 − 0.609i)14-s + (0.565 + 0.652i)15-s + (−0.0287 + 0.0404i)16-s + (0.0704 + 1.47i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.777 - 0.629i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.777 - 0.629i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.230999 + 0.652244i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.230999 + 0.652244i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.327 + 0.945i)T \) |
| 7 | \( 1 + (2.28 + 1.32i)T \) |
| 23 | \( 1 + (4.76 + 0.545i)T \) |
good | 2 | \( 1 + (-0.212 - 0.873i)T + (-1.77 + 0.916i)T^{2} \) |
| 5 | \( 1 + (3.10 - 1.24i)T + (3.61 - 3.45i)T^{2} \) |
| 11 | \( 1 + (0.745 - 3.07i)T + (-9.77 - 5.04i)T^{2} \) |
| 13 | \( 1 + (1.43 - 3.13i)T + (-8.51 - 9.82i)T^{2} \) |
| 17 | \( 1 + (-0.290 - 6.10i)T + (-16.9 + 1.61i)T^{2} \) |
| 19 | \( 1 + (0.0594 - 1.24i)T + (-18.9 - 1.80i)T^{2} \) |
| 29 | \( 1 + (-5.94 + 3.82i)T + (12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (9.45 - 1.82i)T + (28.7 - 11.5i)T^{2} \) |
| 37 | \( 1 + (-5.18 + 4.07i)T + (8.72 - 35.9i)T^{2} \) |
| 41 | \( 1 + (0.438 - 3.05i)T + (-39.3 - 11.5i)T^{2} \) |
| 43 | \( 1 + (3.34 - 3.86i)T + (-6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 + (-0.523 - 0.906i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (13.9 + 1.32i)T + (52.0 + 10.0i)T^{2} \) |
| 59 | \( 1 + (6.84 + 9.60i)T + (-19.2 + 55.7i)T^{2} \) |
| 61 | \( 1 + (-4.24 + 12.2i)T + (-47.9 - 37.7i)T^{2} \) |
| 67 | \( 1 + (5.08 - 4.85i)T + (3.18 - 66.9i)T^{2} \) |
| 71 | \( 1 + (-0.228 - 0.0672i)T + (59.7 + 38.3i)T^{2} \) |
| 73 | \( 1 + (-8.90 + 4.59i)T + (42.3 - 59.4i)T^{2} \) |
| 79 | \( 1 + (-6.53 + 0.623i)T + (77.5 - 14.9i)T^{2} \) |
| 83 | \( 1 + (-1.21 - 8.45i)T + (-79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (1.22 + 0.236i)T + (82.6 + 33.0i)T^{2} \) |
| 97 | \( 1 + (1.14 - 7.97i)T + (-93.0 - 27.3i)T^{2} \) |
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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
−11.30328439778068780576434336843, −10.63696458614780483075362631614, −9.686325330135567189219848603734, −7.985024523224280220448629679251, −7.64965012381879525723663474064, −6.67529871041345798048925625337, −6.25481089905147848559264065534, −4.64004207638485672162206626233, −3.59771383574253408311290705468, −2.01910058187747509907923353808,
0.38464345668556610449822453628, 2.88438055371558740166058058558, 3.47107565908641119411743084468, 4.62857528836091365539825512543, 5.77760740959068428145484068664, 7.09768902793218101249983579726, 7.941580677508477418662256665042, 8.907607703836386342367541160914, 9.921649628050688055957687418740, 10.85823863331210632309938065500