| L(s) = 1 | − 4.14·3-s + 1.70i·7-s + 8.18·9-s + (−3.98 + 10.2i)11-s − 16.6i·13-s + 0.512i·17-s + 19.5i·19-s − 7.07i·21-s − 11.2·23-s + 3.35·27-s + 48.1i·29-s + 5.40·31-s + (16.5 − 42.5i)33-s − 0.530·37-s + 68.8i·39-s + ⋯ |
| L(s) = 1 | − 1.38·3-s + 0.243i·7-s + 0.909·9-s + (−0.362 + 0.931i)11-s − 1.27i·13-s + 0.0301i·17-s + 1.02i·19-s − 0.336i·21-s − 0.488·23-s + 0.124·27-s + 1.65i·29-s + 0.174·31-s + (0.501 − 1.28i)33-s − 0.0143·37-s + 1.76i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.362 + 0.931i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1100 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.362 + 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.3524942440\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3524942440\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + (3.98 - 10.2i)T \) |
| good | 3 | \( 1 + 4.14T + 9T^{2} \) |
| 7 | \( 1 - 1.70iT - 49T^{2} \) |
| 13 | \( 1 + 16.6iT - 169T^{2} \) |
| 17 | \( 1 - 0.512iT - 289T^{2} \) |
| 19 | \( 1 - 19.5iT - 361T^{2} \) |
| 23 | \( 1 + 11.2T + 529T^{2} \) |
| 29 | \( 1 - 48.1iT - 841T^{2} \) |
| 31 | \( 1 - 5.40T + 961T^{2} \) |
| 37 | \( 1 + 0.530T + 1.36e3T^{2} \) |
| 41 | \( 1 + 28.0iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 3.65iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 3.58T + 2.20e3T^{2} \) |
| 53 | \( 1 + 51.9T + 2.80e3T^{2} \) |
| 59 | \( 1 - 41.1T + 3.48e3T^{2} \) |
| 61 | \( 1 + 42.3iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 73.5T + 4.48e3T^{2} \) |
| 71 | \( 1 + 13.3T + 5.04e3T^{2} \) |
| 73 | \( 1 + 107. iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 15.6iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 16.3iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 140.T + 7.92e3T^{2} \) |
| 97 | \( 1 + 97.0T + 9.40e3T^{2} \) |
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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
−9.695663985518748174474648837732, −8.507798041774795100333491878347, −7.61815710067849573609326681270, −6.78846458870204216863101217017, −5.77191183019293594703827423261, −5.34916444479404457320940153274, −4.39662150519443875352888936717, −3.08060461123879557047935883085, −1.59841397260220946223752708067, −0.16388353087832937462928093933,
0.955935007056223401763183098639, 2.52964465027538461652446078047, 4.02784150109427864017103479248, 4.80999302143289094091002145997, 5.78734811689667796058904228335, 6.39210496049214873407657127367, 7.18648138296611952146783778292, 8.255559144416438707346295699732, 9.193369553054679255533829331166, 10.09805152346878778495346796486
