L(s) = 1 | + 1.90i·2-s + (−2.89 − 0.798i)3-s + 0.361·4-s − 0.0951i·5-s + (1.52 − 5.51i)6-s − 10.1·7-s + 8.31i·8-s + (7.72 + 4.61i)9-s + 0.181·10-s − 21.4i·11-s + (−1.04 − 0.288i)12-s − 18.5·13-s − 19.4i·14-s + (−0.0759 + 0.275i)15-s − 14.4·16-s − 11.7i·17-s + ⋯ |
L(s) = 1 | + 0.953i·2-s + (−0.963 − 0.266i)3-s + 0.0904·4-s − 0.0190i·5-s + (0.253 − 0.919i)6-s − 1.45·7-s + 1.03i·8-s + (0.858 + 0.512i)9-s + 0.0181·10-s − 1.95i·11-s + (−0.0872 − 0.0240i)12-s − 1.42·13-s − 1.38i·14-s + (−0.00506 + 0.0183i)15-s − 0.901·16-s − 0.693i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 + 0.963i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.266 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.124823 - 0.163948i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.124823 - 0.163948i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (2.89 + 0.798i)T \) |
| 59 | \( 1 - 7.68iT \) |
good | 2 | \( 1 - 1.90iT - 4T^{2} \) |
| 5 | \( 1 + 0.0951iT - 25T^{2} \) |
| 7 | \( 1 + 10.1T + 49T^{2} \) |
| 11 | \( 1 + 21.4iT - 121T^{2} \) |
| 13 | \( 1 + 18.5T + 169T^{2} \) |
| 17 | \( 1 + 11.7iT - 289T^{2} \) |
| 19 | \( 1 + 16.0T + 361T^{2} \) |
| 23 | \( 1 + 4.05iT - 529T^{2} \) |
| 29 | \( 1 + 13.2iT - 841T^{2} \) |
| 31 | \( 1 + 10.0T + 961T^{2} \) |
| 37 | \( 1 + 26.6T + 1.36e3T^{2} \) |
| 41 | \( 1 - 42.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 55.1T + 1.84e3T^{2} \) |
| 47 | \( 1 - 37.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 72.8iT - 2.80e3T^{2} \) |
| 61 | \( 1 + 69.0T + 3.72e3T^{2} \) |
| 67 | \( 1 - 60.8T + 4.48e3T^{2} \) |
| 71 | \( 1 + 50.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 29.1T + 5.32e3T^{2} \) |
| 79 | \( 1 - 59.8T + 6.24e3T^{2} \) |
| 83 | \( 1 + 92.7iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 130. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 68.0T + 9.40e3T^{2} \) |
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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
−12.19238198854431941177691104370, −11.22959866087519419534994307091, −10.29143871388592307793119427940, −8.992613946280903790177092172389, −7.67012109297102517817085009132, −6.61746668998908275617717956153, −6.10891722408415797670826748163, −4.98995987793898946799763993474, −2.90595205766143857948790908884, −0.12655215050720986348108349976,
2.10174698888677841685914247541, 3.71790680433399748090253913650, 4.97407567528802970604104204933, 6.69136825875944731828397061522, 7.03522128823804514604788294736, 9.428941004416670804230599103019, 10.05065949939282733081930456263, 10.58618610592533508007512312576, 11.97576489305172458840976893367, 12.59200014940716121317148292707