L(s) = 1 | + (1.98 − 0.209i)2-s + 1.73i·3-s + (3.91 − 0.833i)4-s − 4.86i·5-s + (0.362 + 3.44i)6-s + 11.9i·7-s + (7.60 − 2.47i)8-s − 2.99·9-s + (−1.01 − 9.66i)10-s + (1.44 + 10.9i)11-s + (1.44 + 6.77i)12-s + 20.4·13-s + (2.50 + 23.8i)14-s + 8.41·15-s + (14.6 − 6.52i)16-s − 26.3i·17-s + ⋯ |
L(s) = 1 | + (0.994 − 0.104i)2-s + 0.577i·3-s + (0.978 − 0.208i)4-s − 0.972i·5-s + (0.0604 + 0.574i)6-s + 1.71i·7-s + (0.950 − 0.309i)8-s − 0.333·9-s + (−0.101 − 0.966i)10-s + (0.130 + 0.991i)11-s + (0.120 + 0.564i)12-s + 1.57·13-s + (0.179 + 1.70i)14-s + 0.561·15-s + (0.913 − 0.407i)16-s − 1.54i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 264 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.902 - 0.431i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 264 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.902 - 0.431i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.99857 + 0.680205i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.99857 + 0.680205i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.98 + 0.209i)T \) |
| 3 | \( 1 - 1.73iT \) |
| 11 | \( 1 + (-1.44 - 10.9i)T \) |
good | 5 | \( 1 + 4.86iT - 25T^{2} \) |
| 7 | \( 1 - 11.9iT - 49T^{2} \) |
| 13 | \( 1 - 20.4T + 169T^{2} \) |
| 17 | \( 1 + 26.3iT - 289T^{2} \) |
| 19 | \( 1 - 12.5T + 361T^{2} \) |
| 23 | \( 1 + 35.3T + 529T^{2} \) |
| 29 | \( 1 + 13.2T + 841T^{2} \) |
| 31 | \( 1 + 19.6T + 961T^{2} \) |
| 37 | \( 1 - 9.27iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 13.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 23.0T + 1.84e3T^{2} \) |
| 47 | \( 1 + 63.7T + 2.20e3T^{2} \) |
| 53 | \( 1 + 21.7iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 66.8iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 50.9T + 3.72e3T^{2} \) |
| 67 | \( 1 + 87.6iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 13.1T + 5.04e3T^{2} \) |
| 73 | \( 1 - 47.7iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 39.2iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 60.8T + 6.88e3T^{2} \) |
| 89 | \( 1 - 25.2T + 7.92e3T^{2} \) |
| 97 | \( 1 - 107.T + 9.40e3T^{2} \) |
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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
−11.89751428264807424555237427539, −11.30941099406985816547219492645, −9.795744808659072658714714674721, −9.077677244397750077877739449503, −7.993923609737516896970276689449, −6.36364475123699846877665186356, −5.39271955833466699118386179151, −4.73816947986262970774898893620, −3.35236740932827319759347145487, −1.88788375720237814857654522709,
1.45633814889725968994228600975, 3.41655430206515744042203182545, 3.92294279163346187479340503505, 5.91591267823991532025541611652, 6.49427193496033477488552381252, 7.47972628505848687467822195415, 8.301996141437233812941547075961, 10.39398029488445109337822291742, 10.84330068682814946517711559142, 11.58281341257186310622701524983