L(s) = 1 | + 0.472i·2-s + (−1.33 + 2.68i)3-s + 3.77·4-s + 7.63i·5-s + (−1.27 − 0.628i)6-s − 9.20·7-s + 3.67i·8-s + (−5.46 − 7.15i)9-s − 3.61·10-s − 2.06i·11-s + (−5.02 + 10.1i)12-s + 15.4·13-s − 4.35i·14-s + (−20.5 − 10.1i)15-s + 13.3·16-s − 15.6i·17-s + ⋯ |
L(s) = 1 | + 0.236i·2-s + (−0.443 + 0.896i)3-s + 0.944·4-s + 1.52i·5-s + (−0.211 − 0.104i)6-s − 1.31·7-s + 0.459i·8-s + (−0.606 − 0.794i)9-s − 0.361·10-s − 0.187i·11-s + (−0.418 + 0.846i)12-s + 1.18·13-s − 0.310i·14-s + (−1.36 − 0.677i)15-s + 0.835·16-s − 0.921i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.896 - 0.443i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.896 - 0.443i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.276901 + 1.18442i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.276901 + 1.18442i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.33 - 2.68i)T \) |
| 59 | \( 1 + 7.68iT \) |
good | 2 | \( 1 - 0.472iT - 4T^{2} \) |
| 5 | \( 1 - 7.63iT - 25T^{2} \) |
| 7 | \( 1 + 9.20T + 49T^{2} \) |
| 11 | \( 1 + 2.06iT - 121T^{2} \) |
| 13 | \( 1 - 15.4T + 169T^{2} \) |
| 17 | \( 1 + 15.6iT - 289T^{2} \) |
| 19 | \( 1 + 31.9T + 361T^{2} \) |
| 23 | \( 1 - 41.9iT - 529T^{2} \) |
| 29 | \( 1 - 41.7iT - 841T^{2} \) |
| 31 | \( 1 + 6.91T + 961T^{2} \) |
| 37 | \( 1 - 63.0T + 1.36e3T^{2} \) |
| 41 | \( 1 + 28.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 59.9T + 1.84e3T^{2} \) |
| 47 | \( 1 - 37.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 0.0188iT - 2.80e3T^{2} \) |
| 61 | \( 1 + 21.9T + 3.72e3T^{2} \) |
| 67 | \( 1 - 25.9T + 4.48e3T^{2} \) |
| 71 | \( 1 - 40.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 42.2T + 5.32e3T^{2} \) |
| 79 | \( 1 - 32.2T + 6.24e3T^{2} \) |
| 83 | \( 1 + 19.0iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 77.7iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 123.T + 9.40e3T^{2} \) |
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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
−12.78971170830545680770202130454, −11.39243391585098507272706853924, −10.95763381667937682856969214367, −10.17553173640992861856028974172, −9.089694051239487607643402733213, −7.34045804139117231839057179067, −6.39918656834608385935635783407, −5.89065697586729417822490326579, −3.66275522178063576004575429645, −2.86101600310500669062557593379,
0.74592230720582518964709234095, 2.26835838947699783922192194171, 4.19440087409026690479558000657, 6.19174551545272609254534515929, 6.27243028650125139217338778452, 7.967684207470289375916480237009, 8.818262972279586532660158387508, 10.24554779430369391398113281931, 11.19995500546271338464021117874, 12.42932915950689862805487322474